text:
User talk:Takawira~enwikiversity

 Hello Takawira, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 09:59, 1 September 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Takawira. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Takawira~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:44, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:23, 19 April 2015 (UTC)

title: User talk:Takawira~enwikiversity


text:
User talk:Lmnieves

 Hello Lmnieves, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 10:01, 1 September 2010 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:Lmnieves


text:
User talk:Dreanmore989

 Hello Dreanmore989, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 10:01, 1 September 2010 (UTC)

title: User talk:Dreanmore989


text:
User talk:Egm6321.f10.team5.steinberg

 Hello Egm6321.f10.team5.steinberg, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 10:02, 1 September 2010 (UTC)

HW1
Testing

title: User talk:Egm6321.f10.team5.steinberg


text:
User talk:Bryanjeon

 Hello Bryanjeon, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 10:02, 1 September 2010 (UTC)

title: User talk:Bryanjeon


text:
User talk:Egm6321.f10.team06.calle

 Hello Egm6321.f10.team06.calle, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 10:02, 1 September 2010 (UTC)

title: User talk:Egm6321.f10.team06.calle


text:
User talk:Egm6321.fall10.team1.lang

 Hello Egm6321.fall10.team1.lang, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 10:03, 1 September 2010 (UTC)

title: User talk:Egm6321.fall10.team1.lang


text:
User talk:Gccartledge

 Hello Gccartledge, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 10:03, 1 September 2010 (UTC)

 Hi! My name is WikiversityJack. I am a pumpkin and a Wikiversity mascot. My aunt told me that fun + learning = Wikiversity. Let me know if I can help you out.

Motivation and emotion
Hi. Welcome to the unit and to Wikiversity! I look forward to learning with you and I hope you find the topic and the unit rewarding. Please feel free to let me know if you have any suggestions or would like help at any point along the way. Sincerely, James Neill, -- Jtneill - Talk - c 14:50, 2 September 2010 (UTC)

title: User talk:Gccartledge


text:
User talk:Edgar100382

 Hello Edgar100382, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 10:03, 1 September 2010 (UTC)

title: User talk:Edgar100382


text:
User talk:Solblomma5

 Hello Solblomma5, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Harrypotter 10:04, 1 September 2010 (UTC)

title: User talk:Solblomma5


text:
User:Aditya2012

my name is aditya ,studying B.H.M.S. SECOND YEAR in india.

my aim is to aquire and share knowledge with wiki.

title: User:Aditya2012


text:
User:Egm6321.f10.team3.cook

Testing
$$\left( {x}^{a} \right)$$

Team Wiki

Home Work Sandbox
user:egm6321.f10.team3.cook/HW1

user:egm6321.f10.team3.cook/HW2

user:egm6321.f10.team3.cook/HW4

user:egm6321.f10.team3.cook/HW5

user:egm6321.f10.team3.cook/HW6

user:egm6321.f10.team3.cook/HW7

title: User:Egm6321.f10.team3.cook


text:
User:Tomwsulcer

Welcome
Welcome to my user page. Feel free to leave comments on my user talk page -- click on the word discuss, above, or click here; better yet, write something on my Wikipedia talk page which I check much more often.

My free photos at Wikimedia Commons
Click here to see my latest photos. I generally declare my photos to be public domain.

Jakk's Journey
In my spare time I wrote a science fantasy novel, Jakk's Journey, about a high school boy who builds a spaceship, flies to Betelgeuse, meets sexy aliens, and learns how to become a human (the theme). It is available on Amazon in paperback and on Kindle.

My contributions at Wikipedia
Major revamps. I revamped highly trafficked articles:

United States Congress -- about 5000 readers per day; 2800/day April 2012
 * History of the United States Congress about 100/day; 80/day April 2012
 * U.S. Congress and citizens -- 15/day

Wall Street -- 2500/day.

Dating -- 1500/day. But this one still needs rewriting, revamping, more references. Too much advice in it; trimmed it. In December 2011, readership around 2500/day. 1800/day April 2012.

Hope Diamond -- 1500/day; occasionally spikes into the tens of thousands.

Friend zone -- expansion 2500/day

Handyman -- about 160/day; 200/day April 2012

Statistics New Zealand -- 50/day, revamp a while back

USS Iowa (BB-4) -- a US battleship which saw action in the Spanish American War -- 100/day

Citizenship in the United States -- 600 readers per day; floated this article and it's become a mainstay but it probably needs work sometime soon but I don't know what. 400+ April 2012.

Rodney Bingenheimer -- Interesting personality. 300 readers per day (but spikes at times). 90/day April 2012.

Two-party system -- 750 readers per day. 500+ April 2012.

Lake Erie -- first time I've done a lake. 1000 readers per day. But highly important for me to do a lake. Why? I'm no longer a lake virgin. 850+ April.

Man cave -- 300 readers per day. 200+ April 2012.

Kitchen cabinet -- 600/day; this was mostly a copyedit; didn't find much research on it; not exactly a university level subject. 350/day April 2012.

Allegheny College -- 135 readers per day -- revamp a year or so ago; 150/day April 2012.

Union County College -- 35 readers per day, holding steady.

Filter bubble -- 40/day. Floated this article recently. 150/day in April.

Equality of outcome -- 100/day. Expansion.

Equality of autonomy -- floated, hybrid concept developed by Amartya Sen. Lucky if there's 20/day.

Equal opportunity -- one of the major equalities -- 800/day during weekdays. Have you noticed how the equal opportunity gets huge readership while equality of outcome gets much less? Seems grossly unfair that readership is not equal. Hey let's lodge a complaint with the Equal Opportunity Commission.

Rachid Taha -- 300/day? Revamp. I still have no clue what genre this guy is; number of genres is about 10 I think.

Overlook Hospital -- hospital in New Jersey. 20/day

polyethylene#Uses|PEX PEX -- section revamp, 450+day

Ice bath -- brrrrr, floated this article. growing to 125/day -- what's so cool (sorry) about Wikipedia is people read this stuff. Growing to 200+ day, now 350+ a day in May 2012. One of my faster growing ones in terms of readership. 450+/day in June 2012.

Pseudoexfoliation syndrome -- major expansion; eye ailment. 40/day. 60+day. 100+day in May 2012.

Evertune -- newly invented gizmo to keep guitar strings in tune using springs and levers. I want one someday on my guitar.

Ebert test -- can a computer-synthesized voice have enough inflection to make people laugh? Floated; maybe 125/day? 70+day in May 2012.

Community Exchange System -- an alternative bill-less coin-less monetary system working over the Internet which originated in South Africa. About 40/day? 15/day May.

Lake Keowee, a man-made lake in South Carolina -- 50/day

Second Constitutional Convention of the United States -- proposal by small cadre of top academics (usually in law) for substantive political reform 40/day

College admissions in the United States -- 80/day April but hopefully will grow; revamp done May 3 2012. Regularly 250+, sometimes over 300 as of June 2012.
 * Yield (college admissions) -- 10/day
 * Wait list -- 10/day
 * Transfer admissions in the United States -- no idea how this one will go over -- 80/day in early winter 2013
 * FAFSA position -- probably 10/day, colleges data-mine it to screen applicants without their knowledge

Partnership at Drugfree.org -- you know, a nonprofit advertising agency to unsell drugs. Remember the egg in the frying pan commercial? This is your brain. This is drugs. This is your brain on drugs. Revamp. Hopefully this one will get 40 to 50 per day. 140/day in late April; probably some kind of promotion by PDFA.

Lobbying in the United States -- 400/day maybe, not about tennis; that's "Lobbing in the United States"

History of citizenship -- no idea how well this one will do; just floated in February 2012; so far 60/day; 40/day in April; pushing 400/day in November 2012

Jon Wiener -- cool bio of a guy who can sing "I fought the law and I won", unlike most everybody else -- hopefully 30/day after the dust settles

Science 2.0 -- wondering what kind of viewership this one will get; I'm guessing 35/day.

Authority control -- term in library science, rewrite, 400/day. Would be a cool name for a music group.

Medical tricorder -- possible device being built based on the science fiction TV show Star Trek. Probably 40/day

Big History -- cool course and academic discipline. Not the history of obesity but rather the Big Bang to the present. Hopefully 100-200/day. Also I revamped David Christian (historian), founder of the Big History movement, about 60/day.

Judicial interpretation -- about 40/day

Corporate anniversary -- wrote this one; probably 40/day if lucky

Utah Data Center -- varies; much interest in April 2013; I rewrote sections and added references plus did the diagram. Surprisingly, this kind of solution to terrorism is something I wrote about in my book Common Sense II, but it is only one part.

Foodscaping -- not veal parmigiana slithering off the plate to escape, but turning your front lawn into your farm. I'm guessing 10/day, if that.

Biographies. I like to write about folks with unusual occupations. If you have ideas of people to create Wikipedia articles for, please click talk:Tomwsulcer here and write them at the bottom of my user talk page. Biographies I've worked on: Evi Gkotzaridis, Bill Cunliffe, Julian Hatton, Zoya Phan, Heather Mac Donald, Karl Kirchwey, Priscilla Martel, Peter Kapetan, Nate Lee, Sara Nelson, Karyn Marshall, Gar Waterman, Dana Delany, Peter Currie, Sloane Citron, Gary Lee (journalist); Charles T. Howard; Arnold Hiatt; Randy Wayne (biologist); many more. Ones I am considering writing include: Earl Killian, Sara Wedeman, Peter Rysavy, Bill Lewis, Joe Malone.

Wikiproject United States. I'm working with folks on this project.

My interests. Lots of stuff. Sorry, don't know much.

Music. I've worked on articles such as David Wilcox (American musician), Jonatha Brooke, Mountain Man (music group, Offbeats (music group), Elaine Christy, and others. I play guitar.

Handyman stuff. I've been fingerwagged by folks wondering why I write about what I know nothing about and not what I do know about like handyman stuff. Why am I writing about Lake Erie -- they say: you've never been to Lake Erie (true; never been). Why not write about mortar? Sheesh. Okay, okay. I'm trying to contribute more pictures, tips and tool information. Maybe it will keep some homeowner from chopping off a finger.

My free POV writings
These are my POV-oriented writings which contain stuff that generally doesn't belong in Wikipedia. Sorry, people, but Knol is shutting down in May so I'll have to find new venues for them. They can help humans do stuff. They are free documents, generally, with a few exceptions. Photos will perhaps be added soon. I ported some documents to Wordpress or Participedia but haven't figured out how to add photos. Again, if interested, email me and I'll send a text copy to you.

 * Mentally healthy mind (ie positive psychology) This was my hottest knol getting about 150 readers each week, about 9000 until Knol shut down. It's advice about life. I really wish I had known this stuff when I was 21.

 * The Second Constitution of the United States This rewrite of the US Constitution has (1) superior foreign policy architecture (2) citizenship defined (3) federalism restored (4) many other goodies such as privacy, removal of gerrymandering, term limits for congresspersons (& Supreme Court justices too!). It needs intelligent criticism from sharp thinkers, of course, but I think it's loads better than our current United States Constitution. It is a public domain document -- just as well, since I lifted huge chunks from the existing US Constitution.

 * History of citizenship in the United States This began on Wikipedia, moved to Citizendium, moved to Google knol, and is now in Participedia. It's my best guess about why Americans don't participate in politics, and what caused this. Since working on articles in Wikipedia such as History of citizenship, I continue to learn more about this topic. It can also be seen History of citizenship in the United States on Wordpress.

 * How to prevent violence A strategy to prevent crime and violence and terrorism like the Boston Marathon bombing, the Newtown Connecticut shootings, Virginia Tech shootings, Washington DC snipers, Unabomber attacks and pretty much all violence, and rethinks privacy, rights, citizenship in a general way.

 * How to install wire closet shelving -- Home improvement advice.

 * Why women are beautiful (problem is, haven't figured this out yet). So, I didn't write it. But I am bumping into biological and psychological aspects of physical attractiveness; for example, I have a theory about why symmetrical features are often correlated with beauty. Since it is my guess, or as Wikipedians say, "original research", I can not publish it here but I still think that I'm right. Here is my hypothesis about this subject, if interested.

 * Dating and mating in the twenty tens Frank advice for heterosexual men and women seeking love.

 * An American's perspective on New Zealand -- Travel account.

 * Behavioral Economics -- overview of this subject.

 * Philosophy of Spinoza An introduction. Snooze. Boring. Zzzzzzz. Not my most popular piece.

 * Fifteenth Reunion an original screenplay with mild sexual and language content; younger readers please avoid. A romantic comedy. It would be cool if a high school or college drama group did my screenplay. I give permission for groups to use it for small-scale not-for-profit productions, if interested.

 * Ideas on improving Wikipedia. I may write this sooner or later.

 * Amazing humans: Karyn Marshall Interesting story about Karyn Marshall. Weightlifter + chiropractor + woman => champion ==> cool human.

 * Screenplay saga "Americaiad". I'm working on this one too. Themes => America + Spinoza + Aeneid. Heroine is "Kate", an Irish woman who travels Odysseus-like to get back to her family in America and help write the Constitution. That sort of thing. She'll have a guitar bow for singing as well as letting arrows fly at the bad guys! On hold right now.

 * Screenplay "Polar Planet". Outline written. Sexual science fiction. I mapped out the idea, but when I started writing it, it sounded boring and formulaic; so right now it's on hold unless I can make it feel fresh somehow. Ditched it.

 * Jakk's Journey. Science fiction novel about a high schooler who flies to Betelgeuse and meets sexy aliens! Should be fun. Hopefully finished by June. In May 2014, it is on the third draft.

 * Stair Repair -- A handyman project fixing a stair. It's a public domain document, including pictures; feel free to use them for handyman-related articles.

 * Handyman project: mortaring bricks to seal out rain Pictures and text are public domain; feel free to use them for handyman-related articles if interested.

 * Wheeled Wonders A handyman project to de-clutter a packed basement by using movable shelf-carts and psychology.

 * Swapping out a bathroom sink Handyman project on a leaky bathroom sink

 * Keeping raccoons from re-entering a house Yes, this was a recent handyman project which had me learning all about those nocturnal bandit-like creatures. I met a different animal instead.

Interests
I am interested in different subjects and am open to collaborating with people to learn new stuff if they leave comments on my talk page. Stuff I'm interested in: beauty, philosophy, love, being human, psychology, building stuff, simple handyman-level math (easy on the equations), statistics, marketing, buildings, history, politics, and other stuff which I am clueless about, that is, pretty much everything.

title: User:Tomwsulcer


text:
User talk:Tomwsulcer

Feel free to leave comments on this page
I don't know how often I'll be monitoring this page; if you need to reach me, probably best to email me at thomaswrightsulcer@yahoo.com or perhaps leave a comment on one of my Google knols.--Tomwsulcer 11:26, 1 September 2010 (UTC)

title: User talk:Tomwsulcer


text:
User:CuckaOccurs

CuckaOccurs CuckaOccurs 8:36, 1 September 2010 (UTC)

title: User:CuckaOccurs


text:
Counseling/Anatomy of the Brain and Neurosystem

Central nervous system, or CNS

 * skull
 * spine

Peripheral nervous system, or PNS

 * Outside skull and spine
 * Two divisions, external and internal

PNS nerves project from spinal cord to organs (with exceptions)

Somatic nervous system (SNS)
Relates to external environment

nerves

 * afferent -- sensory signals from:
 * skin
 * skeletal muscles
 * joints
 * eye, ears

 * efferent -- motor signals to:
 * skeletal muscles

Autonomic nervous system (ANS)
Regulates internal environment

nerves
Afferent and Efferent

afferent nerves

 * carry sensory signals from internal organs to CNS

efferent nerves

 * carry motor signals from CNS to organs

two kinds:
 * 1) sympathetic
 * 2) parasympathetic

functions:
 * each organ, or autonomic target, receives opposing sympathetic and parasympathetic input
 * organ is controlled by level of sympathetic and parasympathetic activity

sympathetic
 * arousal
 * threatening situations
 * stimulate
 * organize
 * mobilize

parasympathetic
 * relaxation
 * conserve energy

location:
 * sympathetic project from CNS to lumbar and thoratic (chest) regions
 * parasympathetic project from CNS to brain and sacral (lower back) regions

nature: both have two stage neural paths that synapse on other neurons between organs and spinal cord
 * sympathetic synapse farther from organs
 * parasympathetic synapse nearer to organs

cranial nerves
12 parasympathetic cranial nerves project from brain

purely sensory:
 * olafactory
 * optic

sensory and motor:
 * vagus nerves -- gut

disease diagnosis:
 * disruption of nerves can pinpoint tumors and other pathologies

CNS protection
CNS tissue covering
 * meninges -- three membranes that cover CNS (meninx, singular)

cerebrospinal fluid (CSF) -- CNS cushion and physical support, single connected reservoir
 * sub-arachnoid space -- between inner and middle meninges
 * central canal -- runs length of spinal cord
 * ventricles -- four large internal chambers
 * 2 lateral
 * 3rd and 4th vertical

blood brain barrier:
 * tightly-packed blood vessels
 * filters passage of molecules by type
 * filters differently by location (sex hormones)

Neurons
electrochemical signals:
 * reception
 * conduction
 * transmission

external

 * membrane
 * dendrites
 * axon
 * axon hillock
 * myelin
 * nodes of ranveir
 * buttons -- ends of axons
 * synapses

internal
common to all cells:
 * endoplasmic reticulum -- folded membranes
 * ribosomes (rough area) -- synthesis of proteins
 * smooth -- synthesis of fats (without ribosomes)
 * cytoplasm -- clear internal fluid
 * golgi complex -- packages molecules in vesticules (makes vesticules?)
 * micro tubules -- transport tubes

neural transmission

 * synaptic vesticules -- store neurotransmitters
 * neurotransmitters -- chemicals that influence post-synaptic cells

cell membrane

 * lipid bilayer -- membrane surfaces, inside and out
 * thru-membrane passages
 * channel proteins -- allow specific molecules to pass
 * signal proteins -- transfer signal from outside to inside when molecule binds on outside

classes of neurons

 * multipolar (most common) -- two or more processes (axons/dendrites) extending from body
 * unipolar -- one process extending from body that can divide into multiple extentions
 * bipolar -- two processes extending from body
 * interneurons -- short, or no, axons conduct signals within a single brain structure

gross neural structures
nuclei -- clusters of cell bodies in CNS ganglia -- clusters of axons in PNS

bundles of axons:
 * tracts in CNS
 * nerves in PNS

glial cells
90% of brain cells, primarily support neurons

 * oligodendrocytes -- CNS
 * create myelin sheaths around
 * wrap around axons in CNS sending out extensions (does it stay connected, or release myelin?)
 * schwann cells -- PNS
 * create myelin sheaths
 * creates only one sheath and can regenerate myelin
 * astrocytes -- some in blood barrier system
 * transport chemicals from blood to neuron bodies
 * extensions cover blood vessels
 * connect to neuron bodies
 * micoglia -- respond to injury or disease
 * engulf cellular debris
 * trigger inflammatory responses

axis
anterior / posterior dorsal / ventral
 * front / back
 * front of brain, but upwards in body
 * top / bottom
 * dorsal is top of brain, but back of body
 * ventral is bottom of brain, but front of body

superior -- top

inferior -- bottom

in / out

 * medial -- towards midline
 * lateral -- away from midline towards lateral, or outside, surfaces

far / close
distal -- far from CNS proximal -- close to CNS

sections

 * sagittal -- vertically divides left and right
 * frontal -- vertically divides front and back
 * horizontal -- divides top and bottom
 * cross section -- cut across nerve or spinal cord

gray matter

 * core
 * H-shaped
 * cell bodies
 * unmyelated neurons
 * horns
 * dorsal
 * ventral

white matter

 * surrounding
 * myelated axons (white lipid)

attached nerves

 * 31 pairs of nerves attached
 * left and right (bundles of axons)
 * axons go to ventral and dorsal horns
 * sensory neurons (unipolar) go to dorsal horns with bodies outside in nerve bundle forming dorsal root ganglia
 * motor axons emerge from ventral with neuron bodies in horn
 * somatic project to the skeletal system
 * autonomic connect to ganglia (clusters of cell bodies) that synapse on neurons that connect to organs

Brain
5 divisions from top to bottom:
 * telenephalon
 * diencephalon
 * mesencephalon
 * metencephalon
 * myenlencephalon (medulla)

spinal cord is beneath myenlencephalon

From 3 sections:
 * forebrain, telenephalon and diencephalon
 * midbrain, mesencephalon
 * hindbrain, metencephalon and myenlencephalon

myenlencephalon

 * tracts for signals between brain and body
 * also called medulla

rectiliniar formation
spans:
 * network of 100 tiny nuclei within central core of brain stem
 * posterior boundary of myelencephalon (bottom of medulla)
 * anterior boundary of mesencephalon (top of midbrain)

rectiliniar activation system:
 * sleep
 * attention
 * movement
 * muscle tone maintenance
 * heart beat
 * circulation
 * respiratory reflexes

diencephalon
thalamus and hypothalmus

thalamus

 * 2 lobes joined by massa intermedia
 * top of brainstem
 * on top of third ventricle

 * pairs of nuclei that project to cortex (most)

 * sensory relay nuclei (many)
 * receive signals from sensory receptors
 * process signals
 * transmit to specific areas of sensory cortex

lateral geniculate nuclei visual sensory relay nuclei

hypothalamus

 * below (hypo) thalamus
 * regulation of motivated behaviors
 * regulates release of hormones from pituitary gland (attached below/front)

attached structures:
 * pituitary gland
 * optic chasm
 * decussate or contralateral (cross over to other side)
 * ipsilateral (staying on same side)
 * mammilary bodies
 * behind pituitary

telencephalon

 * largest division
 * voluntary movement initiation
 * interprets (processes) sensory movement
 * mediates complex cognitive processes
 * learning
 * speaking
 * problem solving

cerebral cortex

also cortex
 * layer of tissue
 * convoluted, or furrowed
 * increase amount of cerebral cortex from same tissue mass

convolutions:
 * fissures, large furrows
 * singular suclcus, small furrows
 * gyri, ridges between fissures

longitudinal fissure:
 * separates hemispheres
 * largest fissure
 * connections between hemispheres (tracts)
 * cerebral commisures
 * largest, corpus callosum
 * minimal number of tracts

lateral fissure: major horizontal fissure

four lobes divided by horizontal and latteral fissures:
 * frontal
 * parietal
 * temporal
 * occipital

major gyri:
 * precentral, contain motor cortex
 * postcentral, somatosensory cortex (body sensation)
 * superior temporal gyri, auditory cortex

neocortex:
 * 6-layered cortex
 * 90% of cortex
 * evolutionary recent
 * numbered I through VI

two types of neurons:

pyramidal
 * multipolar
 * large
 * apex runs to surface of cortex
 * long axon

stellate
 * interneurons
 * star-shaped
 * small with short or no axon

non-neo cortex

 * fewer layers

 * hippocampus
 * memory
 * medial edge of cerebral cortex
 * curves back on itself in medial temporal lobe

 * cingulate cortex

limbic system
circuit of mid-line structures in brain

motivated behaviors members:
 * fleeing
 * feeding
 * fighting
 * sex
 * hippocampus
 * cingulate cortex

sub-cortical
 * mamillary bodies
 * amygdala
 * fornix
 * septum

basal ganglia

 * voluntary movement
 * sub-cortial structures

members:
 * amygdala
 * caudate (circle)
 * putamen (within circle, straitum with caudate)
 * globus pallidus


text:
User:A Hock

Hi, my name is Aaron Hockaday. I'm currently doing my e-portfolio as a blog, which can be found here: http://letsgetmotivatedandemotional.blogspot.com

title: User:A Hock


text:
User talk:A Hock

 Hello A Hock, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. Jtneill - Talk - c 13:49, 1 September 2010 (UTC)

 Hi , I am Percy. Nice to meet you. I love the beach, swimming, fishing, and I am currently doing some research at Wikiversity on oceanography. I am also learning to play piano. What do you like? Can I be your mascot?

Motivation and emotion
Hi. Welcome to the unit and to Wikiversity! I look forward to learning with you and I hope you find the topic and the unit rewarding. Please feel free to let me know if you have any suggestions or would like help at any point along the way. Sincerely, James Neill, -- Jtneill - Talk - c 13:49, 1 September 2010 (UTC)

-- Jtneill - Talk - c 13:25, 15 December 2010 (UTC)

title: User talk:A Hock


text:
User:Nayantaka

nayantaka is my nickname while my real name is dwi budi santoso. Currently i manage a blog named : komunitasblog.info, it's an educational blog.

title: User:Nayantaka


text:
Motivation and emotion/Textbook/Preface

Welcome to this free and open textbook for the psychological study of motivation and emotion.

Motivation and emotion are core topics in undergraduate psychology teaching around the world. This book focuses on applied theory and research topics in motivation and emotion. Chapter foci and content were initially developed by undergraduate psychology participants at the University of Canberra in consultation with the lecturer/editor, James Neill during the second half of 2010. Whilst the book may be useful as a whole, each individual chapter is also designed to stand alone.

This first edition was developed through a participant-authored, instructor-edited, open wiki collaboration learning and assessment exercise during Semester 2, 2010 (August-November). Approximately 80 participants enrolled initially and three months later 70 submitted chapters. Participants created Wikiversity accounts of their own preference, thus some used real names and some used other account names. Participants learnt how to use Wikiversity through demonstrations and through developing individual e-portfolios (worth 30% of the unit). Each participant selected or negotiated a textbook chapter topic (50%) and made a 5 minute multimedia recording of his/her textbook chapters (20%).

The plan for the future is to continue developing the quality of content on current topics and to develop new topics in collaboration with new participant cohorts. In this way, it is hoped to develop a high-quality open textbook which can supplement and eventually replace the reliance on commercial textbooks. At the very least it is hoped that this textbook project provides a practical demonstration and example of the potential for the development of open textbooks as a learning and assessment exercise in higher education. This is also an example of open academia in action. {{Hide in print|

title: Motivation and emotion/Textbook/Preface


text: Motivation and emotion/Textbook/Front cover


text:
Motivation and emotion/Textbook/Banner

Motivation and emotion (2010) A free and open psychology textbook authoring exercise by students

title: Motivation and emotion/Textbook/Banner


text:
Open Space Technology

Open Space Technology (OST) is an approach for hosting meetings, conferences, corporate-style retreats and community summit events, focused on a specific and important purpose or task—but beginning without any formal agenda, beyond the overall purpose or theme.

Law of Mobility
If at any time you find yourself in any situation where you are neither learning nor contributing: go someplace else.

Self-Organization
Highly scalable and adaptable, OST has been used in meetings of 5 to 2,100 people. The approach is characterized by five basic mechanisms:
 * 1) a broad, open invitation that articulates the purpose of the meeting;
 * 2) participant chairs arranged in a circle;
 * 3) a "bulletin board" of issues and opportunities posted by participants;
 * 4) a "marketplace" with many breakout spaces that participants move freely between, learning and contributing as they "shop" for information and ideas;
 * 5) a "breathing" or "pulsation" pattern of flow, between plenary and small-group breakout sessions.

The approach is most distinctive for its initial lack of an agenda, which sets the stage for the meeting's participants to create the agenda for themselves, in the first 30–90 minutes of the meeting or event. Typically, an Open Space meeting will begin with short introductions by the sponsor (the official or acknowledged leader of the group) and usually a single facilitator. The sponsor introduces the purpose; the facilitator explains the "self-organizing" process called "Open Space." Then the group creates the working agenda, as individuals post their issues in bulletin board style. Each individual "convener" of a breakout session takes responsibility for naming the issue, posting it on the bulletin board, assigning it a space and time to meet, and then later, showing up at that space and time, kicking off the conversation, and taking notes. These notes are usually compiled into a proceedings document that is distributed physically or electronically to all participants. Sometimes one or more additional approaches are used to sort through the notes, assign priorities and identify what actions should be taken next. Throughout the process, the ideal facilitator is described as being "fully present and totally invisible" (see Owen, User's Guide), "holding a space" for participants to self-organize, rather than managing or directing the conversations.

Hundreds of open space meetings have been documented (http://www.openspaceworld.org; Open Space Institute US, STORIES Newsletter; http://www.openspaceworldscape.org; Tales from Open Space, edited by Harrison Owen, Abbott Publishing). In "Open Space Technology: A User's Guide," (and seven other books about Open Space), Harrison Owen explains that this approach works best when four conditions are present, namely high levels of (1) complexity, in term of the tasks to be done or outcomes achieved; (2) diversity, in terms of the people involved and/or needed to make any solution work; (3) real or potential conflict, meaning people really care about the central issue or purpose; and (4) urgency, meaning that the time to act was "yesterday".

According to Harrison Owen, originator of the term and the approach, Open Space works because it harnesses and acknowledges the power of self-organization, which he suggests is substantially aligned with the deepest process of life itself, as described by leading-edge complexity science as well as ancient spiritual teachings.

Origin and Ownership
The history of Open Space Technology is detailed in the Introduction to "Open Space Technology: A User's Guide", by Harrison Owen. (Paperback: 192 pages, Berrett-Koehler Publishers; 3 edition (April 1, 2008), Language: English, ISBN 1576754766, ISBN 978-1576754764)

In the early 1980s, Harrison Owen wrote a paper on what he called "Organization Transformation". He presented this paper at a traditional management conference. It was well enough received that a number of people urged Owen to organize a conference to specifically address the issues and opportunities he identified in his paper. Owen hosted the first annual Symposium on Organization Transformation in 1983, in a traditional conference format, in Monterrey, California. The event was a success, inasmuch as it was generally agreed that it should happen again. The second annual symposium (OT-2) one year later, but still in a traditional conference format.

Harrison Owen agreed to organize OT-3 for the following year, but by his own account, did not relish another year of work to manage all the details. Upon volunteering to host the third symposium, he retreated to the bar, where he consistently claims to have discovered what he later called the "Open Space" approach to meetings and events, at the bottom of his second martini. His plan for the following year's symposium was informed by his experience as a biblical scholar, associate pastor, peace corps organizer in the villages of west Africa, and federal government staffer and organization development consultant in Washington DC.

The following year, he sent out a simple, one-paragraph invitation, and more than 100 people showed up to discuss Organization Transformation. In his main meeting room he set the chairs one large circle and proceeded to explain that what participants could see in the room was the extent of his organizing work. If they had an issue or opportunity that they felt passionate about and wanted to discuss with other participants, they should come to the center of the circle, get a marker and paper, write their issue and their name, read that out, and post it on the wall. It took about 90 minutes for the 100+ people to organize a 3-day agenda of conference sessions, each one titled, hosted, and scheduled by somebody in the group.

Participants at OT-1 and OT-2 said that the best part of the events was the coffee breaks, which Owen always pointed out was the one part of the event that he didn't plan and couldn't take credit for. His inspiration to articulate the theme, the larger purpose for the work of the symposium, in an invitation and then a brief opening comment, and then simply "open the space" for participants to self-organize around the issues and opportunities they saw as essential to that purpose, was a conscious decision to make "more of what works". His martini-based plan sought to minimize the grunt work by leadership (him) and assign responsibility for maximizing productive learning and contribution to his participants (everyone else).

The approach worked well, in the 3-1/2 days symposium, where it was repeated annually through OT-20. Soon after the first "open space" event at OT-3, however, Owen tried the same approach with a consulting client, a large chemical firm and a group of polymer chemists. When it worked there, too, the participants of OT began trying it out with their clients, in a variety of different kinds of organizations, to address many different kinds of strategic and community issues, in countries around the world. They returned to the OT symposium each year to share learnings.

Owen never trademarked or patented or certified "Open Space" in any way. He always claimed to have discovered, rather than invented, it. He said it could be practiced freely by anyone with a good head and good heart. From the beginning, he said only that those who used the approach and found it valuable, should share their stories and learnings as freely, as well.

Twenty-five years later, Harrison Owen estimates that more than 100,000 different "open space" meetings have taken place. The Open Space World Map (http://www.openspaceworldmap.org) documents that these events have taken place in more than 160 countries. In December 2009, the OSLIST email listserve (hosted by Boise State University at http://listserv.boisestate.edu/cgi-bin/wa?A0=OSLIST) for practitioners worldwide had 660+ members in 39 countries and more than 26,500 publicly searchable messages, relating to all aspects of practice. Information about Open Space is now posted in 21 different languages at Open Space World (http://www.openspaceworld.org). There are at least five different government-chartered associations or institutes (France, Germany, Portugal, Sweden and USA) promoting Open Space practice around the world, and also active, but informal, organizations in several other countries (including Canada, Taiwan, Australia and New Zealand, and the UK). The German-language Yahoo group started February 2002, had 233 members at year-end 2009, mostly from Germany, Austria and Switzerland and also from France, Spain, The Netherlands, Poland and elsewhere, with 3497 messages in its archive. At year-end 2009, the Australian email group was more than 500 strong.

Harrison Owen originally used the term "open space" for his "self-organizing meetings". One of the earliest implementations of the approach was for a conference theme of "The business of business is learning," in Goa, India. The organizer of the conference was interviewed by the local media and described the simple process. When asked what the process was called, he embellished it a bit, with the more important sounding "Open Space Technology". The story was picked up by the New York Times (need date, c. 1985), and so "Open Space" became "Open Space Technology".

Ideal Initial Conditions
According to Open Space Technology: A User's Guide and other books by Harrison Owen, Open Space Technology works best when four conditions are present:

 * 1) a high level of complexity, such that no single person or small group fully understands or can solve the issue
 * 2) a high level of diversity, in terms of the skills and people required for a successful resolution
 * 3) real or potential conflict, which implies that people genuinely care about the issue
 * 4) a high level urgency,  meaning the time for decisions and action was "yesterday"

Further, the recognition of these conditions by leadership typically implies some level of letting go of control and opening of invitation. In different ways and to varying degrees, leaders convening Open Space meetings acknowledge that they, personally, do not have "the answer" to whatever complex, urgent and important issue(s) must be addressed and they put out the call (invitation) to anyone in the organization or community who cares enough to attend a meeting and try to create a solution.

Typical Meeting Process
At the beginning of an Open Space the participants sit in a circle, or in concentric circles for large groups (300 to 2000 people and more).

The facilitator will greet the people and briefly re-state the theme of their gathering, without giving a lengthy speech. Then someone will invite all participants to identify any issue or opportunity related to the theme. Participants willing to raise a topic will come to the centre of the circle, write it on a sheet of paper and announce it to the group before choosing a time and a place for discussion and posting it on a wall. That wall becomes the agenda for the meeting.

No participant must suggest issues, but anyone may do so. However, if someone posts a topic, the system expects that the person has a real passion for the issue and can start the discussion on it. That person also must make sure that a report of the discussion is done and posted on another wall so that any participant can access the content of the discussion at all times. No limit exists on the number of issues that the meeting can post.

When all issues have been posted, participants sign up and attend those individual sessions. Sessions typically last for 1.5 hours; the whole gathering usually lasts from a half day up to about two days. The opening and agenda creation lasts about an hour, even with a very large group.

After the opening and agenda creation, the individual groups go to work. The attendees organize each session; people may freely decide which session they want to attend, and may switch to another one at any time. Online networking can occur both before and following the actual face-to-face meetings so discussions can continue seamlessly. All discussion reports are compiled in a document on site and sent to participants, unedited, shortly after.

In this way, Open Space Technology begins without any pre-determined agenda, but work is directed by a "theme" or "purpose" or "invitation" that is carefully articulated by leaders, in advance of the meeting. The organizers do outline in advance a schedule of breakout times and spaces. The combination of clear purpose and ample breakout facilities directly supports the process of self-organization by meeting participants. After the opening briefing, the facilitator typically remains largely in the background, exerting no control over meeting content or participants, though possibly supporting the compiling of whatever sort of document is produced by participants.

Small groups might create agendas of only a few issues. Very large groups have generated as many as 234 sessions running concurrently over the course of a day and longer meetings may establish priorities and set up working-groups for follow-up.

Guiding Principles and One Law
In his User's Guide, Harrison Owen has articulated "Four Principles" and "One Law" that are typically quoted and briefly explained during the opening briefing of an Open Space meeting. These explanations describe rather than control the process of the meeting. The four principles and Owen's explanations are:

 * 1) Whoever comes is the right people ...reminds participants that they don't need the CEO and 100 people to get something done, you need people who care.  And, absent the direction or control exerted in a traditional meeting, that's who shows up in the various breakout sessions of an open space meeting.
 * 2) Whenever it starts is the right time ...reminds participants that "spirit and creativity do not run on the clock."
 * 3) Whatever happens is the only thing that could have ...reminds participants that once something has happened, it's done—and no amount of fretting, complaining or otherwise rehashing can change that.  Move on.
 * 4) When it's over, it's over ...reminds participants that we never know how long it will take to resolve an issue, once raised, but that whenever the issue or work or conversation is finished, move on to the next thing.  Don't keep rehashing just because there's 30 minutes left in the session.  Do the work, not the time.

Owen explains his one "Law," called the "Law of Two Feet" or "The Law of Mobility", as follows: If at any time during our time together you find yourself in any situation where you are neither learning nor contributing, use your two feet, go someplace else. In this way, all participants are given both the right and the responsibility to maximize their own learning and contribution, which the Law assumes only they, themselves, can ultimately judge and control. When participants lose interest and get bored in a breakout session, or accomplish and share all that they can, the charge is to move on, the "polite" thing to do is go something else. In practical terms, Owen explains, the Law of Two Feet says: "Don't waste time!"

title: Open Space Technology


text: Template:Motivation and emotion/Textbook chapter


text:
User:EGM6321.f10.team1.russo

This is my Principles of Engineering Analysis 1 wiki page. Below are the contributions that I made to my team's assignments. I took this course Fall semester of 2010.

Given
given the boundary values:

$$\begin{align} & y(a)=\alpha \\ & y(b)=\beta \\ \end{align}$$.

Find
Solve for variables c and d in the following equation:

 * {| style="width:100%" border="0"

 * style="width:95%" |
 * style="width:95%" |

$$y(x)=cy_{H}^{1}(x)+dy_{H}^{2}(x)+{{y}_{p}}(x)$$

(p5-3)
 *
 * }

Solution
Substituting the boundary point data into the given equation yields:

 * {| style="width:100%" border="0"

$$y(a)= \alpha = cy_{H}^{1}(a)+dy_{H}^{2}(a)+{{y}_{p}}(a)$$ (4.1),(4.2)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

 * {| style="width:100%" border="0"

$$y(b)= \beta = cy_{H}^{1}(b)+dy_{H}^{2}(b)+{{y}_{p}}(b)$$ (4.3),(4.4)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Using algebra to solve equations (4.2) and (4.4) for variable c yields:

 * {| style="width:100%" border="0"

$$c = \frac{1}{y_{H}^{1}(a)}[\alpha - dy_{H}^{2}(a)-{{y}_{p}}(a)]$$ (4.5)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

 * {| style="width:100%" border="0"

$$c = \frac{1}{y_{H}^{1}(b)}[\beta - dy_{H}^{2}(b)-{{y}_{p}}(b)]$$ (4.6)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Setting equations (4.5) and (4.6) equal to one another by recognizing that c=c yields:

 * {| style="width:100%" border="0"

$$\frac{1}{y_{H}^{1}(a)}[\alpha - dy_{H}^{2}(a)-{{y}_{p}}(a)]= \frac{1}{y_{H}^{1}(b)}[\beta - dy_{H}^{2}(b)-{{y}_{p}}(b)]$$ (4.7)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

Solving equation (4.7) for variable d using algebra:

 * {| style="width:100%" border="0"

$$\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}[\alpha - dy_{H}^{2}(a)-{{y}_{p}}(a)]= \beta - dy_{H}^{2}(b)-{{y}_{p}}(b)$$ (4.8)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

 * {| style="width:100%" border="0"

$$\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}[\alpha -{{y}_{p}}(a)]+{{y}_{p}}(b)-\beta = \frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}dy_{H}^{2}(a) - dy_{H}^{2}(b) = d(\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}y_{H}^{2}(a) - y_{H}^{2}(b))$$ (4.9),(4.10)
 * style="width:95%" |
 * style="width:95%" |
 *
 * }

 * {| style="width:100%" border="0"

$$d = \frac{\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}[\alpha -{{y}_{p}}(a)]+{{y}_{p}}(b)-\beta}{\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}y_{H}^{2}(a) - y_{H}^{2}(b)}$$ (4.11)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Equation (4.11) is the final answer for variable d.

Now, variable c may be found by substituting equation (4.11) into equation (4.5):

 * {| style="width:100%" border="0"

$$c = \frac{1}{y_{H}^{1}(a)}[\alpha - \frac{\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}[\alpha -{{y}_{p}}(a)]+{{y}_{p}}(b)-\beta}{\frac{y_{H}^{1}(b)}{y_{H}^{1}(a)}y_{H}^{2}(a) - y_{H}^{2}(b)}y_{H}^{2}(a)-{{y}_{p}}(a)]$$ (4.12)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Equation (4.12) is the final answer for variable c, and so the solution is now complete.

EGM6321.f10.team1.russo 22:23, 13 September 2010 (UTC)

Given

 * {| style="width:100%" border="0"

$$ \varnothing(x,y)= x^2y^\frac{3}{2}+log(x^3y^2)=k$$ (4.1)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Find
1)$$ F(x,y,y')=\frac{d\varnothing}{dx}(x,y)$$ 2) Verify that F is exact N1_ODE 3) Invent three more N1_ODE's

Solution
This problem asks for F(x,y,y') and for the verification that F is exactly N1_ODE. In other words, verify that there exists a function $$ \phi(x,y)\ $$ such that:
 * {| style="width:100%" border="0"

$$ F = \frac{d\phi}{dx}$$ (4.2)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

In other terms, verify that:

 * {| style="width:100%" border="0"

$$F(x,y,y') = \frac{d}{dx}\phi(x,y) = \frac{d}{dx}k$$ (4.3, 4.4)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Solving for the partial derivatives in equations 2 and 3 yields:

 * {| style="width:100%" border="0"

$$F(x,y,y') = \frac{d\phi}{dx}(x,y) + \frac{d\phi}{dx}(x,y)\frac{dy}{dx} = M(x,y) + N(x,y)\frac{dy}{dx} = 0$$ (4.5, 4.6 and 4.7)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Where:

 * {| style="width:100%" border="0"

$$M(x,y) = \frac{d\phi}{dx}(x,y) = (2)(y^{3/2})x + \frac{3}{x}$$ (4.8, 4.9)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

 * {| style="width:100%" border="0"

$$N(x,y) = \frac{d\phi}{dx}(x,y) = \frac{3}{2}(x^2)(y^{1/2})x + \frac{2}{y}$$ (4.10, 4.11)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

1) and 2) Substituting equations 4.9 and 4.11 into equation 4.5 yields the final answer for F(x,y,y') (4.12) and also verifies that it is exactly N1_ODE (4.13)

 * {| style="width:100%" border="0"

$$F(x,y,y') = ((2)(y^{3/2})x + \frac{3}{x}) + (\frac{3}{2}(x^2)(y^{1/2})x + \frac{2}{y})y' = 0$$ (4.12, 4.13)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

3) Below are 3 examples of N1_ODEs

 * {| style="width:100%" border="0"

$$ F(x,y,y') = 22x^2y + y^5y'\ $$ (4.15)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$ F(x,y,y') = sin(x^{7/2}) - \frac{4}{3}y^6y'\ $$ (4.16)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }
 * {| style="width:100%" border="0"

$$ F(x,y,y') = x^8 + y^4y'\ $$ (4.17)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Author: User:EGM6321.f10.team1.russo

Find
Use the integrating factor method to obtain the result of eq3.1 directly

Solution
The given solution is one of a first order linear ODE. The derivative of such an equation has the form:

Where $$\ \tau\, $$ is the constant of integration. We will assume the case of constant coefficients, so this equation becomes:

Equation 3.3 can be rearranged as:

We desire a solution for $$\ x(\tau\,) $$, which may be found using the integration by factors method. First, the function $$\ h(\tau\,) $$ must be found:

Multiplying both sides of equation 3.4 by $$\ h(\tau\,) $$ yields:

This equation may be simplified by recalling that $$\ h(\tau\,)a(\tau\,) = \dot{h}(\tau\,) $$:

Substituting equation 3.5 into equation 3.7 for $$h(\tau\,)$$ yields:

It is now possible to find an expression for x(t) by integrating both sides of this equation with respect to $$\ \tau\, $$:

Performing this integration yields:

By performing some algebra on this equation, x(t) is found:

Author: EGM6321.f10.team1.russo 07:15, 6 October 2010 (UTC)

Find
Find $$ y_{xxxxx}\ $$ in terms of the derivative of y with respect to t

Solution
Author Egm6321.f10.team1.allison 17:08, 21 October 2010 (UTC), EGM6321.f10.team1.russo 16:31, 2 November 2010 (UTC)

Given
and

Find
Show that eq 6.1 agrees with with expression in King, page 8, eq 1.6:

Where:

Solution
Equation (6.2) may be rearranged as follows:

Taking the derivative of both sides of equation (6.5) yields:

Carrying out the derivative of the left side of equation (6.6) yields:

Upon inspection, one can notice a relationship between equations (6.7) and (6.4), which may be written as:

It is now possible to develop an expression for $$ h(x)\ $$ by setting equations (6.6) and (6.8) equal to one another and rearranging:

King's expression (equation (6.3)) may be rearranged, and its integration variables substituted, yielding: Where the two homogeneous solutions, namely $$ Au_1(x)\ $$ and $$ Bu_2(x)\ $$ have been omitted so that only the particular solution is considered.

The objective now is to use rearrangement and substitution to put the particular solution in a form equivalent to equation (6.10). The rule of Integration by Parts may be written as follows:

Rearranging this rule into an applicable form yields:

It can easily be seen that the left side of equation (6.12) is of nearly the same form as the particular solution, equation (6.1).

To apply the Integration by Parts rule to the particular solution, the following substitutions may be made: and

Substituting equations (6.13),(6.14) and (6.15) into equation (6.12) yields:

Multiplying both sides of equation (6.15) by $$ u_1(x)\ $$ and substituting equation (6.9) for $$ h(x)\ $$ in the right side of (6.16) yields:

By inspection, the left side of equation (6.17) is exactly the particular solution from equation (6.1). Thus, after rearranging the right side of equation (6.17) and canceling out like terms, the particular solution may be written as:

It is my belief that the given particular solution from lecture 30-1 equation 2 is a misprint. If the particular equation had been given as:

Then equation (6.18) would instead have become:

Which is equivalent to King's expression for y(x) as stated in equation (6.10).

Author EGM6321.f10.team1.russo 00:58, 3 November 2010 (UTC)

Given
Where $$\ [n/2] $$ = the integer part of $$\ n/2 $$

Find
Verify that Eq8.3 thru Eq8.7 can be written in the form of Eq8.1 or Eq8.2

Solution
Because equations 8.1 and 8.2 are equivalent, it is only necessary to prove that equations 8.3 through 8.7 may be written in the form of equation 8.1.

1) Verify that equation 8.3 may be written in the form of equation 8.1.

Equation 8.3 is given as:

By observing the subscript in equation 8.3, the two required parameters may be found: and

Equation 8.1 now becomes:

Performing the summation in equation 8.10 yields:

After performing the necessary calculations, equation 8.11 becomes:

This expression of equation 8.1 is equivalent to equation 8.3.

Therefore, equation 8.3 may be written as equation 8.1

2) Verify that equation 8.4 may be written in the form of equation 8.1.

Equation 8.4 is given as:

By observing the subscript in equation 8.4, the two required parameters may be found: and

Equation 8.1 now becomes:

Performing the summation in equation 8.15 yields:

After performing the necessary calculations, equation 8.16 becomes:

This expression of equation 8.1 is equivalent to equation 8.4.

Therefore, equation 8.4 may be written as equation 8.1

3) Verify that equation 8.5 may be written in the form of equation 8.1.

Equation 8.5 is given as:

By observing the subscript in equation 8.5, the two required parameters may be found: and

Equation 8.1 now becomes:

Performing the summation in equation 8.20 yields:

After performing the necessary calculations, equation 8.21 becomes:

This expression of equation 8.1 is equivalent to equation 8.5.

Therefore, equation 8.5 may be written as equation 8.1

4) Verify that equation 8.6 may be written in the form of equation 8.1.

Equation 8.6 is given as:

By observing the subscript in equation 8.6, the two required parameters may be found: and

Equation 8.1 now becomes:

Performing the summation in equation 8.25 yields:

After performing the necessary calculations, equation 8.26 becomes:

This expression of equation 8.1 is equivalent to equation 8.6.

Therefore, equation 8.6 may be written as equation 8.1

5) Verify that equation 8.7 may be written in the form of equation 8.1.

Equation 8.7 is given as:

By observing the subscript in equation 8.7, the two required parameters may be found: and

Equation 8.1 now becomes:

Performing the summation in equation 8.30 yields:

After performing the necessary calculations, equation 8.31 becomes:

This expression of equation 8.1 is equivalent to equation 8.7.

Therefore, equation 8.7 may be written as equation 8.1

It has thus been verified that equations 8.3 through 8.7 may be written as such.

Therefore, equations 8.3 through 8.7 may be written in the form of equation 8.1 or 8.2

Author EGM6321.f10.team1.russo 02:40, 12 November 2010 (UTC)

Find
Verify that Eq9.3 thru Eq9.7 are solutions of Legendre Equations as given in Eq9.1 or Eq9.2

Solution
Since equations 9.2 and 9.1 are identical, it is only necessary to prove that equations 9.3 through 9.7 are solutions to equation 9.1. This will be accomplished by taking the first and second derivatives of each potential solution equation, substituting the results into equation 9.1, and then verifying that there exists an integer $$\ n $$ for which the Legendre differential equation is satisfied:

1) Verify that equation 9.3 is a solution of equation 9.1:

Begin with the potential solution:

Taking the first derivative of equation 9.3 yields:

and taking the second derivative of equation 9.3 yields:

Substituting $$\ P_0(x)$$, $$\ P_0'(x)$$, and $$\ P_0''(x) $$ into equation 9.1 yields:

Carrying out the multiplication yields:

It can be seen that there are two values of the parameter n that satisfy this condition.

When $$\ n=0 $$, equation 9.11 becomes:

And when $$\ n=-1 $$ equation 9.11 becomes:

when $$\ n=0 $$ or when $$\ n=-1 $$

2) Verify that equation 9.4 is a solution of equation 9.1:

Begin with the potential solution:

Taking the first derivative of equation 9.4 yields:

and taking the second derivative of equation 9.4 yields:

Substituting $$\ P_1(x)$$, $$\ P_1'(x)$$, and $$\ P_1''(x) $$ into equation 9.1 yields:

Carrying out the multiplication yields:

Dividing both sides of equation 9.18 by $$\ x $$ and rearranging the remaining terms yields: It can be seen that there are two values of the parameter n that satisfy this condition.

When $$\ n=1 $$, equation 9.19 becomes:

And when $$\ n=-2 $$ equation 9.19 becomes:

when $$\ n=1 $$ or when $$\ n=-2 $$

3) Verify that equation 9.5 is a solution of equation 9.1:

Begin with the potential solution:

Taking the first derivative of equation 9.5 yields:

and taking the second derivative of equation 9.5 yields:

Substituting $$\ P_2(x)$$, $$\ P_2'(x)$$, and $$\ P_2''(x) $$ into equation 9.1 yields:

Carrying out the multiplication yields:

Rearranging equation 9.26 yields:

Dividing both sides of equation 9.27 by $$\ \frac{1}{2}(3x^2 - 1) $$ yields: It can be seen that there are two values of the parameter n that satisfy this condition.

When $$\ n=2 $$, equation 9.28 becomes:

And when $$\ n=-3 $$ equation 9.28 becomes:

when $$\ n=2 $$ or when $$\ n=-3 $$

4) Verify that equation 9.6 is a solution of equation 9.1:

Begin with the potential solution:

Taking the first derivative of equation 9.6 yields:

and taking the second derivative of equation 9.6 yields:

Substituting $$\ P_3(x)$$, $$\ P_3'(x)$$, and $$\ P_3''(x) $$ into equation 9.1 yields:

Carrying out the multiplication yields:

Rearranging equation 9.35 yields:

Dividing both sides of equation 9.36 by $$\ \frac{1}{2}(5x^3 - 3x) $$ yields: It can be seen that there are two values of the parameter n that satisfy this condition.

When $$\ n=3 $$, equation 9.37 becomes:

And when $$\ n=-4 $$ equation 9.37 becomes:

when $$\ n=3 $$ or when $$\ n=-4 $$

4) Verify that equation 9.7 is a solution of equation 9.1:

Begin with the potential solution:

Taking the first derivative of equation 9.7 yields:

and taking the second derivative of equation 9.7 yields:

Substituting $$\ P_4(x)$$, $$\ P_4'(x)$$, and $$\ P_4''(x) $$ into equation 9.1 yields:

Carrying out the multiplication yields:

Rearranging equation 9.44 yields:

Dividing both sides of equation 9.45 by $$\ \left(\frac{35}{8}x^4 - \frac{15}{4}x^2 + \frac{3}{8}\right) $$ yields: It can be seen that there are two values of the parameter n that satisfy this condition.

When $$\ n=4 $$, equation 9.46 becomes:

And when $$\ n=-5 $$ equation 9.46 becomes:

when $$\ n=4 $$ or when $$\ n=-5 $$

It has thus been verified that equations 9.3 through 9.7 are solutions to equation 9.1.

Therefore, equations 9.3 through 9.7 are valid solutions to Legendre's differential equation as shown in equation 9.1

Author EGM6321.f10.team1.russo 23:08, 11 November 2010 (UTC)

Given
Using either: or:

Find
A) Prove that Eq 3.3 is even for k=0,1,2...

B} Prove that Eq 3.4 is odd for k=0,1,2...

Solution
A) Prove that Equation 3.3 is even for k=0,1,2,... using Equation 3.1:

To prove that Equation 3.3 is even, it must be shown that it fulfills the definition of an even function, which is as follows:

Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all x in the domain of f:

Therefore, it must be shown that the following condition is satisfied:

This will be shown using Equation 3.1. First, it must be put into the proper form. Upon inspection, it can be seen that $$\ n = 2k $$ and that $$\ x = \mu $$ for equation 3.3.

Substituting these parameters into equation 3.1 yields:

Performing some of the algebra in equation 3.6 yields:

If $$ \mu $$ were replaced with $$\ -\mu $$, then equation 3.1 would instead have become:

The only term in equation 3.7 that contains the variable $$\ \mu $$ is $$\ (\mu^2)^{k - i} $$, while the only term in equation 3.8 containing $$\ \mu $$ is $$\ ((-\mu)^2)^{k - i} $$. Since $$\ \mu $$ and $$\ k $$ are integers, $$\ \mu^2 $$ must equal $$\ (-\mu)^2 $$.

This means that 3.7 must equal equation 3.8, and so condition 3.5 is satisfied.

B) Prove that Equation 3.4 is odd for k=0,1,2,... using Equation 3.1:

To prove that Equation 3.4 is odd, it must be shown that it fulfills the definition of an odd function, which is as follows:

Again, let f(x) be a real-valued function of a real variable. Then f is odd if the following equation holds for all x in the domain of f:

Therefore, it must be shown that the following condition is satisfied:

This will be shown using Equation 3.1. First, it must be put into the proper form. Upon inspection, it can be seen that $$\ n = 2k+1 $$ and that $$\ x = \mu $$ for equation 3.4.

Substituting these parameters into equation 3.1 yields:

Performing some of the algebra in equation 3.10 and multiplying both sides by -1 yields:

If $$ \mu $$ were replaced with $$\ -\mu $$, then equation 3.1 would instead have become:

By bringing the negative sign from the term $$\ (-\mu) $$ to the front of equation 3.12, this may be rewritten as:

Recalling the fact that $$\ \mu^2 $$ must equal $$\ (-\mu)^2 $$ as discussed in part A above, equation 3.11 must equal equation 3.13, and so condition 3.9 is satisfied.

The work of Team 1 from the Fall 2009 term was referenced during this solution. Specifically, the team's solution to Problem 1 of Homework 7 was viewed to find out how the definition of an even or odd function corresponds to equation 3.1. This reference may be found at the following link:

While the referenced material simply mentions the correlations between equation 3.1 and the properties of even and odd functions, the above solution begins with these properties and proves that they are followed by equations 3.3 and 3.4.

Author EGM6321.f10.team1.russo 22:54, 22 November 2010 (UTC)

Given
From the solution to Problem 7.8 above, we are given equations 8.7 and 8.8, which are simply renamed below:

And

Find
Continue Power Series expansion of Eq10.1 and use equations 10.4-10.6 to find $$ \left \{ P_3, P_4, P_5, P_6 \right \} \ $$ and compare results with those obtained by a) Solutions from HW6.8

b) Solutions from HW7.9

Solution
$$\begin{align} & \text{The generating function for the Legendre Polynomials was derived during meeting} \\ & \text{40 so therefore we will just present the result as follows:} \\ & \\  & \left[ A\left( \mu ,\rho  \right) \right]^{-\frac{1}{2}}=\frac{1}{\sqrt{1-2\mu \rho +\rho ^{2}}}=\frac{1}{\sqrt{1-x}}\text{              where }x=2\mu \rho -\rho ^{2} \\ & \\  & \text{We would like to expand the generating function as a power series in terms of }\rho \text{ to the} \\ & \text{order 6}\text{. This is done by the following:} \\ & \\  & \left[ A\left( \mu ,\rho  \right) \right]^{-\frac{1}{2}}=\sum\limits_{i=0}^{6}{\alpha _{i}x^{i}}=\alpha _{0}+\alpha _{1}x+\alpha _{2}x^{2}+\alpha _{3}x^{3}+\alpha _{4}x^{4}+\alpha _{5}x^{5}+\alpha _{6}x^{6} \\ & \\  & \text{We can then use the given information, as well as equation 10}\text{.8 to solve for the }\alpha _{i}\text{ }\!\!'\!\!\text{ s}\text{.} \\ & \\  & \alpha _{0}=1 \\ & \alpha _{1}=\frac{1}{2} \\ & \alpha _{2}=\frac{1\cdot 3}{2\cdot 4}=\frac{3}{8} \\ & \alpha _{3}=\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}=\frac{15}{48}=\frac{5}{16} \\ & \alpha _{4}=\frac{1\cdot 3\cdot 5\cdot 7}{2\cdot 4\cdot 6\cdot 8}=\frac{105}{384}=\frac{35}{128} \\ & \alpha _{5}=\frac{1\cdot 3\cdot 5\cdot 7\cdot 9}{2\cdot 4\cdot 6\cdot 8\cdot 10}=\frac{945}{3840}=\frac{63}{256} \\ & \alpha _{6}=\frac{1\cdot 3\cdot 5\cdot 7\cdot 9\cdot 11}{2\cdot 4\cdot 6\cdot 8\cdot 10\cdot 12}=\frac{10395}{46080}=\frac{231}{1024} \\ & \\  & \text{Now we use our definition for }x\text{ to organize the correct }\alpha _{i}'s\text{ with the correct powers of }\rho. \\ & \text{To we use the integer form of the binomial expansion to expand all }x\text{ expressions}\text{. The} \\ & \text{binomial theorem is given as follows:} \\ & \\  & \left( a+b \right)^{n}=\sum\limits_{r=0}^{n}{\frac{n!}{r!\left( n-r \right)!}a^{n-r}b^{r}} \\ & \\  & x=2\mu \rho -\rho ^{2} \\ & \\  & x^{2}=\left( 2\mu \rho -\rho ^{2} \right)^{2}=4\mu ^{2}\rho ^{2}-4\mu \rho ^{3}+\rho ^{4} \\ & \\  & x^{3}=\left( 2\mu \rho -\rho ^{2} \right)^{3}=\sum\limits_{r=0}^{3}{\frac{3!}{r!\left( 3-r \right)!}\left( 2\mu \rho  \right)^{3-r}\left( -\rho ^{2} \right)^{r}} \\ & \text{  }=\frac{3\cdot 2\cdot 1}{1\left( 3\cdot 2\cdot 1 \right)}\cdot 8\mu ^{3}\rho ^{3}-\frac{3\cdot 2\cdot 1}{1\left( 2\cdot 1 \right)}\cdot 4\mu ^{2}\rho ^{4}+\frac{3\cdot 2\cdot 1}{2\cdot 1\left( 1 \right)}\cdot 2\mu \rho ^{5}-\frac{3\cdot 2\cdot 1}{3\cdot 2\cdot 1\left( 1 \right)}\rho ^{6} \\ & \therefore x^{3}=8\mu ^{3}\rho ^{3}-12\mu ^{2}\rho ^{4}+6\mu \rho ^{5}-\rho ^{6} \\ & \\  & x^{4}=\left( 2\mu \rho -\rho ^{2} \right)^{4}=\sum\limits_{r=0}^{4}{\frac{4!}{r!\left( 4-r \right)!}\left( 2\mu \rho  \right)^{4-r}\left( -\rho ^{2} \right)^{r}} \\ & \text{  }=\frac{4\cdot 3\cdot 2\cdot 1}{1\left( 4\cdot 3\cdot 2\cdot 1 \right)}16\mu ^{4}\rho ^{4}-\frac{4\cdot 3\cdot 2\cdot 1}{1\left( 3\cdot 2\cdot 1 \right)}8\mu ^{3}\rho ^{5}+\frac{4\cdot 3\cdot 2\cdot 1}{2\cdot 1\left( 2\cdot 1 \right)}4\mu ^{2}\rho ^{6}-\frac{4\cdot 3\cdot 2\cdot 1}{3\cdot 2\cdot 1\left( 1 \right)}2\mu \rho ^{7}+\frac{4\cdot 3\cdot 2\cdot 1}{4\cdot 3\cdot 2\cdot 1\left( 1 \right)}\rho ^{8} \\ & \text{  }=16\mu ^{4}\rho ^{4}-32\mu ^{3}\rho ^{5}+24\mu ^{2}\rho ^{6}-8\mu \rho ^{7}+\rho ^{8} \\ & \\  & x^{5}=\left( 2\mu \rho -\rho ^{2} \right)^{5}=\sum\limits_{r=0}^{5}{\frac{5!}{r!\left( 5-r \right)!}\left( 2\mu \rho  \right)^{5-r}\left( -\rho ^{2} \right)^{r}} \\ & \text{  }=\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{1\left( 5\cdot 4\cdot 3\cdot 2\cdot 1 \right)}32\mu ^{5}\rho ^{5}-\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{1\left( 4\cdot 3\cdot 2\cdot 1 \right)}16\mu ^{4}\rho ^{6}+\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{2\cdot 1\left( 3\cdot 2\cdot 1 \right)}8\mu ^{3}\rho ^{7}-\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{3\cdot 2\cdot 1\left( 2\cdot 1 \right)}4\mu ^{2}\rho ^{8}+ \\ & \text{     }\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{4\cdot 3\cdot 2\cdot 1\left( 1 \right)}2\mu \rho ^{9}-\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{5\cdot 4\cdot 3\cdot 2\cdot 1\left( 1 \right)}\rho ^{10} \\ & \\  & \therefore x^{5}=32\mu ^{5}\rho ^{5}-80\mu ^{4}\rho ^{6}+80\mu ^{3}\rho ^{7}-40\mu ^{2}\rho ^{8}+10\mu \rho ^{9}-\rho ^{10} \\ & \\  & x^{6}=\left( 2\mu \rho -\rho ^{2} \right)^{6}=\sum\limits_{r=0}^{6}{\frac{6!}{r!\left( 6-r \right)!}\left( 2\mu \rho  \right)^{6-r}\left( -\rho ^{2} \right)^{r}} \\ & \text{  }=\frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{1\left( 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1 \right)}64\mu ^{6}\rho ^{6}-\frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{1\left( 5\cdot 4\cdot 3\cdot 2\cdot 1 \right)}32\mu ^{5}\rho ^{7}+\frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{2\cdot 1\left( 4\cdot 3\cdot 2\cdot 1 \right)}16\mu ^{4}\rho ^{8}-\frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{3\cdot 2\cdot 1\left( 3\cdot 2\cdot 1 \right)}8\mu ^{3}\rho ^{9}+ \\ & \text{   }\frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{4\cdot 3\cdot 2\cdot 1\left( 2\cdot 1 \right)}4\mu ^{2}\rho ^{10}-\frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{5\cdot 4\cdot 3\cdot 2\cdot 1\left( 1 \right)}2\mu \rho ^{11}+\frac{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1\left( 1 \right)}\rho ^{12} \\ & \text{ } \\ & \therefore x^{6}=64\mu ^{6}\rho ^{6}-192\mu ^{5}\rho ^{7}+240\mu ^{4}\rho ^{8}-160\mu ^{3}\rho ^{9}+60\mu ^{2}\rho ^{10}-12\mu \rho ^{11}+\rho ^{12} \\ & \\  & \text{After plugging in the values for the respective }x\text{ }\!\!'\!\!\text{ s and }\alpha \text{ }\!\!'\!\!\text{ s, we group the terms together which have common } \\ & \text{factors of }\rho ^{i}.\text{ Each of these factors represent an order of expansion and their respective coeffiecents are} \\ & \text{Legendre polynomials of their respective order}\text{. We were asked to calculate up to order 6, so after } \\ & \text{rearranging and factoring the expansion looks as follows}\text{.} \\ & \\  & \frac{1}{\sqrt{1-2\mu \rho +\rho ^{2}}}=1+\mu \rho +\frac{1}{2}\left( 3\mu ^{2}-1 \right)\rho ^{2}+\frac{1}{2}\left( 5\mu ^{3}-3\mu  \right)\rho ^{3}+\left( \frac{35}{8}\mu ^{4}-\frac{15}{4}\mu ^{2}+\frac{3}{8} \right)\rho ^{4}+\left( \frac{63}{8}\mu ^{5}-\frac{35}{4}\mu ^{3}+\frac{15}{8}\mu  \right)\rho ^{5} \\ & \text{                       +}\left( \frac{231}{16}\mu ^{6}-\frac{315}{16}\mu ^{4}+\frac{105}{16}\mu ^{2}-\frac{5}{16} \right)\rho ^{6} \\ \end{align}$$

PART A)

In Lecture 36, we were given the following two identical equations:

For homework problem 6.8, the results of these two equations were found for n=[0,4] as follows:

Using equation 10.21 to find $$\ P_5(x) $$ yields:

Likewise, equation 10.21 may be used to find $$\ P_6(x) $$, yielding:

Upon inspection, it is obvious that equation 10.18 is equivalent to 10.26, 10.19 is equivalent to 10.27, 10.20 is equivalent to 10.29, and 10.21 is equivalent to 10.31.

Therefore, the power series expansion of equation 10.1 yields the same results as equations 10.21 and 10.22

PART B)

$$\begin{align} & \text{We can see just by inspection the Legendre Polynomials which were derived by the recurrence } \\ & \text{relation in Problem 6 above, and the Legendre Polynomials which were dervied by power series} \\ & \text{expansion in the first part of Problem 10 here are equal}\text{.} \\ & \text{The Legendre polynomials found in Problem 6 are as follows:} \\ & \\  & P_{3}\left( \mu  \right)=\frac{1}{2}\left( 5\mu ^{3}-3\mu  \right) \\ & P_{4}\left( \mu \right)=\frac{35}{8}\mu ^{4}-\frac{15}{4}\mu ^{2}+\frac{3}{8} \\ & P_{5}\left( \mu \right)=\frac{63}{8}\mu ^{5}-\frac{35}{4}\mu ^{3}+\frac{15}{8}\mu  \\ & P_{6}\left( \mu \right)=\frac{1}{16}\left( 231\mu ^{6}-315\mu ^{4}+105\mu ^{2}-5 \right) \\ & \\  & \text{The Legendre Polynomials found in the first part of Problem 10 are equal to the above}\text{.} \\ \end{align}$$

$$\begin{align} & \text{Because we see that we get the same exact solutions by expanding the generating function by} \\ & \text{power series as we do using the 2 general equations for the Legendre Polynomials, there must be} \\ & \text{a way to go directly from the generating function to the general expression}\text{. We know from the } \\ & \text{above power series expansion the following is true}\text{.} \\ & \\  & \frac{1}{\sqrt{1-2\mu \rho +\rho ^{2}}}=\sum\limits_{n=0}^{\infty }{P_{n}\left( \mu  \right)\rho ^{n}} \\ & \\  & \text{The goal is to seek a path to directly derive a general expression for }P_{n}\left( \mu  \right)\text{ by using the binomial} \\ & \text{expansion on the Legendre generating function}\text{. The binomial expansion is defined as follows:} \\ & \\  & \left( a+b \right)^{n}=\sum\limits_{r=0}^{n}{\frac{n!}{r!\left( n-r \right)!}}a^{n-r}b^{r} \\ & \\  & \text{We know that a factoral is defined for only positive integers, but it }\!\!'\!\!\text{ s obvious if we are to move directly} \\ & \text{from the generating function into an expansion we need to be creative in our approach}\text{. In this light let }\!\!'\!\!\text{ s} \\ & \text{assume for a second that we can have a factorial of a negative non-integer}\text{. If we can manipulate the } \\ & \text{factorial in such a way that in the end, all the undefined characteristics of the factorial; i}\text{.e}\text{. the negative} \\ & \text{non-integer parts are removed, then we have a direct method of expressing the generating function in} \\ & \text{a general sense and should be able to derive the following equation}\text{.} \\ \end{align}$$

$$\begin{align} & \\  & P_{n}\left( \mu  \right)=\sum\limits_{i=0}^{\frac{n}{2}}{\left( -1 \right)^{i}\frac{\left( 2n-2i \right)!x^{n-2i}}{2^{n}i!\left( n-i \right)!\left( n-2i \right)!}} \\ & \\  & \text{Therefore let }\!\!'\!\!\text{ s assume the following is valid and see what pops out}\text{.} \\ \end{align}$$

$$\begin{align} & \\  & \left( 1-x \right)^{-\frac{1}{2}}=\sum\limits_{i}^{\infty }{\frac{\left( -\frac{1}{2} \right)!\left( -1 \right)^{i}}{i!\left( -\frac{1}{2}-i \right)!}x^{i}} \\ \end{align}$$

$$\begin{align} & \\  & \text{We see the problem up front lies in the factor of the binomial coefficient, } \\ & \\  & \frac{\left( -\frac{1}{2} \right)!}{\left( -\frac{1}{2}-i \right)!} \\ & \\ \end{align}$$

$$\begin{align} & \text{We know right away we can deal with the negative number issue by multiplying through by 1}^{i}.\text{ So let }\!\!'\!\!\text{ s } \\ & \text{expand out the first few terms and see if some common factors will pop out of both the numerator and } \\ & \text{the denominator}\text{. Remember, because these expressions do not exist in a sense that there is not a factoral } \\ & \text{definition for them, we can not make any assumption about the tail end of these expressions either in the } \\ & \text{numerator or the denominator}\text{. We must leave them trailing off; i}\text{.e}\text{. they just keep going and going to infinity}\text{.} \\ \end{align}$$

$$\begin{align} & \\  & \frac{\left( -\frac{1}{2} \right)\left( -\frac{1}{2}-1 \right)\left( -\frac{1}{2}-2 \right)\left( -\frac{1}{2}-3 \right)............................................}{\left( -\frac{1}{2}-i \right)\left( -\frac{1}{2}-i-1 \right)\left( -\frac{1}{2}-i-2 \right)\left( -\frac{1}{2}-i-3 \right).........................} \\ & \\  & \text{Now let }\!\!'\!\!\text{ s get rid of }i\text{ number of negative terms from the numerator}\text{.} \\ & \\  & \frac{\left( \frac{-1}{-1} \right)^{i}\left( -\frac{1}{2} \right)\left( -\frac{1}{2}-1 \right)\left( -\frac{1}{2}-2 \right)\left( -\frac{1}{2}-3 \right)............................................}{\left( -\frac{1}{2}-i \right)\left( -\frac{1}{2}-i-1 \right)\left( -\frac{1}{2}-i-2 \right)\left( -\frac{1}{2}-i-3 \right)...................................} \\ & \\  & \frac{\left( -1 \right)^{i}\left( \frac{1}{2} \right)\left( \frac{1}{2}+1 \right)\left( \frac{1}{2}+2 \right)\left( \frac{1}{2}+3 \right)..........\left( \frac{1}{2}+i-1 \right)\left( -\frac{1}{2}-i \right)\left( -\frac{1}{2}-i-1 \right)\left( -\frac{1}{2}-i-2 \right)\left( -\frac{1}{2}-i-3 \right).......}{\left( -\frac{1}{2}-i \right)\left( -\frac{1}{2}-i-1 \right)\left( -\frac{1}{2}-i-2 \right)\left( -\frac{1}{2}-i-3 \right)........................................................................} \\ & \\  & \text{What we find is that the entire denominator (all terms to infinity) divide out all the negative terms} \\ & \text{in the numerator, therefore leaving the fraction free of negative terms spanning all the way to infinity}\text{.} \\ & \text{Even though we know nothing about these terms in the beginning, we know under these manipulations} \\ & \text{they do they are irrelevant to our solution}\text{. } \\ \end{align}$$

$$\begin{align} & \\  & \left( -1 \right)^{i}\left( \frac{1}{2} \right)\left( \frac{1}{2}+1 \right)\left( \frac{1}{2}+2 \right)\left( \frac{1}{2}+3 \right)........\left( \frac{1}{2}+i-1 \right) \\ & \\  & \left( -1 \right)^{i}\left( \frac{1}{2} \right)\left( \frac{3}{2} \right)\left( \frac{5}{2} \right)\left( \frac{7}{2} \right)........\left( \frac{2i-1}{2} \right) \\ & \\  & \frac{\left( -1 \right)^{i}1\cdot 3\cdot 5\cdot 7.........\left( 2i-1 \right)}{2^{i}} \\ & \\  & \text{Because we are looking for a way to expand in terms of the binomial expansion, we want to derive an} \\ & \text{expression that will be factorial in nature}\text{. Therefore we multiply the numerator by the missing even } \\ & \text{numbers}\text{.} \\ & \\  & \frac{\left( -1 \right)^{i}1\cdot 3\cdot 5\cdot 7.........\left( 2i-1 \right)}{2^{i}}\cdot \frac{2\cdot 4\cdot 6........2i}{2\cdot 4\cdot 6........2i} \\ & \\  & \frac{\left( -1 \right)^{i}1\cdot 3\cdot 5\cdot 7.........\left( 2i-1 \right)\cdot \left( 2\cdot 4\cdot 6........2i \right)}{2^{i}\cdot \left( 2\cdot 4\cdot 6........2i \right)} \\ & \\  & \frac{\left( -1 \right)^{i}1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7.........\left( 2i-1 \right)\cdot 2i}{2^{i}\cdot \left[ 2^{i}\left( 1\cdot 2\cdot 3\cdot 4........\text{ }i \right) \right]} \\ & \\  & \frac{\left( -1 \right)^{i}2i\cdot \left( 2i-1 \right).........3\cdot 2\cdot 1}{2^{2i}\left( i \right)!} \\ & \\  & \frac{\left( -1 \right)^{i}\left( 2i \right)!}{2^{2i}i!} \\ & \\  & \text{Therefore, we have successfully derived an expression for our negative, non-integer factorial}\text{.} \\ & \\  & \left( 1-x \right)^{-\frac{1}{2}}=\sum\limits_{i}^{\infty }{\frac{\left( -\frac{1}{2} \right)!\left( -1 \right)^{i}}{i!\left( -\frac{1}{2}-i \right)!}x^{i}}=\sum\limits_{i}^{\infty }{\frac{\left( -1 \right)^{2i}\left( 2i \right)!}{2^{2i}\left( i! \right)^{2}}x^{i}} \\ & \\  & \text{Using this form, it is easy to see that we can apply this to our generating function for }P_{n}\left( \mu  \right). \\ &  \\  & \frac{1}{\sqrt{1-2\mu \rho +\rho ^{2}}}=\frac{1}{\sqrt{1-\left( 2\mu \rho -\rho ^{2} \right)}}=\sum\limits_{i}^{\infty }{\frac{\left( -1 \right)^{2i}\left( 2i \right)!}{2^{2i}\left( i! \right)^{2}}\left( 2\mu \rho -\rho ^{2} \right)^{i}}=\sum\limits_{i}^{\infty }{\frac{\left( -1 \right)^{2i}\left( 2i \right)!}{2^{2i}\left( i! \right)^{2}}\left( \rho \left( 2\mu -\rho  \right) \right)^{i}} \\ & \\  & \sum\limits_{i}^{\infty }{\frac{\left( -1 \right)^{2i}\left( 2i \right)!}{2^{2i}\left( i! \right)^{2}}\left( \rho \left( 2\mu -\rho  \right) \right)^{i}}=\sum\limits_{i}^{\infty }{\frac{\left( -1 \right)^{2i}\left( 2i \right)!}{2^{2i}\left( i! \right)^{2}}\rho ^{i}\left( 2\mu -\rho  \right)^{i}} \\ & \\  & \text{We again want to expand }\left( 2\mu -\rho  \right)^{i}\text{ using the binomial expansion}\text{.} \\ & \\  & \left( 2\mu -\rho  \right)^{i}=\sum\limits_{k=0}^{i}{\frac{i!}{k!\left( i-k \right)!}}\left( 2\mu  \right)^{i-k}\left( -\rho  \right)^{k} \\ & \\  & \sum\limits_{i}^{\infty }{\frac{\left( -1 \right)^{2i}\left( 2i \right)!}{2^{2i}\left( i! \right)^{2}}\rho ^{i}\left( 2\mu -\rho  \right)^{i}}=\sum\limits_{i}^{\infty }{\frac{\left( -1 \right)^{2i}\left( 2i \right)!}{2^{2i}\left( i! \right)^{2}}\rho ^{i}}\sum\limits_{k=0}^{i}{\frac{i!}{k!\left( i-k \right)!}}\left( 2\mu  \right)^{i-k}\left( -\rho  \right)^{k} \\ \end{align}$$

$$\begin{align} & \sum\limits_{i=0}^{\infty }{\sum\limits_{k=0}^{i}{\frac{\left( -1 \right)^{2i}\left( 2i \right)!}{2^{2i}\left( i! \right)^{2}}\rho ^{i}\frac{i!}{k!\left( i-k \right)!}}}\left( 2\mu \right)^{i-k}\left( -\rho  \right)^{k} \\ & \\  & \text{Now we make the following simplifications:} \\ & \left( -1 \right)^{2i}=1 \\ & \left( -\rho \right)^{k}=(-1)^{k}\rho ^{k} \\ & \frac{i!}{\left( i! \right)^{2}}=\frac{1}{i!} \\ & \\  & \text{This leaves us with the following:} \\ & \\  & \sum\limits_{i=0}^{\infty }{\sum\limits_{k=0}^{i}{\frac{\left( 2i \right)!\left( -1 \right)^{k}\left( 2\mu  \right)^{i-k}\rho ^{i+k}}{2^{2i}i!\left( i-k \right)!k!}}} \\ & \\  & \text{Now let }\!\!'\!\!\text{ s look again at our vey first relationship between the generating function and the form of the} \\ & \text{expansion}\text{.} \\ & \\  & \frac{1}{\sqrt{1-2\mu \rho +\rho ^{2}}}=\sum\limits_{n=0}^{\infty }{P_{n}\left( \mu  \right)\rho ^{n}} \\ & \\  & \text{We see that if we are to have our expression match up to the series expansion, we need to make }\rho \text{ } \\ & \text{independent of }k.\text{ We can easily do this by asking ourself, if we let }k\text{ go to a new variable, say n,} \\ & \text{what do we allow }i\text{ to go to, or do we leave }i\text{ alone? The logic is easily seen in the following:} \\ & \\  & \text{If }k\to n \\ & \text{Then }i+n+?=i\text{ } \\ & \text{We see that we let }i\to i-n \\ & \\  & \text{This then allows us to express }\rho \text{ as being independent of the second summation as can be seen}\text{.} \\ & \\  & \sum\limits_{i=0}^{\infty }{\sum\limits_{k\to n=0}^{i\to \left( i-n \right)}{\frac{\left( 2i\to 2\left( i-n \right) \right)!\left( -1 \right)^{k\to n}\left( 2\mu  \right)^{i\to \left( i-n \right)-k\to \left( -n \right)}\rho ^{i\to \left( i-n \right)+k\to n}}{2^{2i\to 2\left( i-n \right)}\left( i\to i-n \right)!\left( i\to \left( i-n \right)-k\to \left( -n \right) \right)!\left( k\to n \right)!}}}=\sum\limits_{i=0}^{\infty }{\left\{ \sum\limits_{n=0}^{i-n}{\frac{\left( 2i-2n \right)!\left( -1 \right)^{n}\left( 2\mu  \right)^{i-2n}}{2^{2i-2n}\left( i-n \right)!\left( i-2n \right)!n!}} \right\}}\rho ^{i} \\ & \\  & \text{This is exactely the form that we are looking for and it can be seen that the upper} \\ & \text{limit of the inner summation be }\frac{i}{2},\text{ not }i-n.\text{ } \\ & \\ \end{align}$$

$$\begin{align} & \text{Therefore, our derivation of the general expression for }P_{n}\left( \mu \right)\text{ is complete and we have shown } \\ & \text{mathematically why on Problem }\!\!\#\!\!\text{ 6 and Problem  }\!\!\#\!\!\text{ 10 Part A above we got the same answer as } \\ & \text{in the beginning of Problem }\!\!\#\!\!\text{ 10}\text{.} \\ \end{align}$$

$$\begin{align} & \text{Proof of }(i-n)\to \frac{i}{2} \\ & \text{Keep LHS variables the same and change RHS variables}\text{. This should allow a relationship to be } \\ & \text{derived on what happens to }i\text{ and }k\text{ when I set them to }i-n\text{ and }n\text{ respectively}\text{.} \\ & \\  & \frac{\left( 2i \right)!(-1)^{k}\left( 2\mu  \right)^{i-k}\rho ^{k+i}}{2^{2i}i!\left( i-k \right)!k!}=\frac{\left( 2i \right)!(-1)^{k}\left( 2\mu  \right)^{i-k}\rho ^{k+i}}{2^{2i}i!\left( i-k \right)!k!} \\ & \\  & \frac{\left( 2 \right)!\left( i \right)!(-1)^{k}\left( 2 \right)^{i-k}\left( \mu  \right)^{i-k}\rho ^{k+i}}{2^{2i}i!\left( i-k \right)!k!}=\frac{\left( 2 \right)!\left( i \right)!(-1)^{k}\left( 2 \right)^{i-k}\left( \mu  \right)^{i-k}\rho ^{k+i}}{2^{2i}i!\left( i-k \right)!k!} \\ & \\  & \rho \text{ will be independent of }P_{n}\left( \mu  \right)\text{ so we can divide it out}\text{.} \\ & \\  & \frac{\left( 2 \right)!\left( i \right)!(-1)^{k}\left( 2 \right)^{i-k}\left( \mu  \right)^{i-k}}{2^{2i}i!\left( i-k \right)!k!}=\frac{\left( 2 \right)!\left( i \right)!(-1)^{k}\left( 2 \right)^{i-k}\left( \mu  \right)^{i-k}}{2^{2i}i!\left( i-k \right)!k!} \\ & \\  & \text{Change LHS variables; i}\text{.e}\text{. let }i\to i-n\text{ and }k\to n. \\ & \\  & \frac{\left( 2 \right)!\left( i \right)!(-1)^{k}\left( 2 \right)^{i-k}\left( \mu  \right)^{i-k}}{2^{2i}i!\left( i-k \right)!k!}=\frac{\left( 2 \right)!\left( i-n \right)!(-1)^{n}\left( 2 \right)^{i-n-n}\left( \mu  \right)^{i-n-n}}{2^{2(i-n)}\left( i-n \right)!\left( i-n-n \right)!n!} \\ & \\  & \frac{\left( 2 \right)!\left( i \right)!(-1)^{k}\left( 2 \right)^{i-k}\left( \mu  \right)^{i-k}}{2^{2i}i!\left( i-k \right)!k!}=\frac{\left( 2 \right)!\left( i-n \right)!(-1)^{n}\left( 2 \right)^{i-2n}\left( \mu  \right)^{i-2n}}{2^{2i-2n}\left( i-n \right)!\left( i-2n \right)!n!} \\ & \\  & \text{Expand expontential terms on RHS and LHS}\text{.} \\ & \\  & \frac{\left( 2 \right)!\left( i \right)!(-1)^{k}\left( 2 \right)^{i}\left( \mu  \right)^{i-k}}{2^{2i}i!\left( i-k \right)!2^{k}k!}=\frac{2^{2n}\left( 2 \right)!\left( i-n \right)!(-1)^{n}\left( 2 \right)^{i}\left( \mu  \right)^{i-2n}}{2^{2i}2^{2n}\left( i-n \right)!\left( i-2n \right)!n!} \\ & \\  & \text{Divide out terms on RHS and LHS}\text{.} \\ & \\  & \frac{\left( \mu  \right)^{i-k}}{2^{i}\left( i-k \right)!2^{k}k!}=\frac{\left( 2 \right)^{i}\left( \mu  \right)^{i-2n}}{2^{2i}\left( i-2n \right)!n!} \\ & \\  & \frac{\left( \mu  \right)^{i-k}}{\left( i-k \right)!2^{k}k!}=\frac{\left( 2 \right)^{2i}\left( \mu  \right)^{i-2n}}{2^{2i}\left( i-2n \right)!n!} \\ & \\  & \frac{\left( \mu  \right)^{i-k}}{\left( i-k \right)!2^{k}}=\frac{\left( \mu  \right)^{i-2n}}{\left( i-2n \right)!} \\ & \\  & \frac{\left( \mu  \right)^{i-k}}{2^{k}\left( i-k \right)!}=\frac{\left( \mu  \right)^{2\left( \frac{i}{2}-n \right)}}{\left( 2\left( \frac{i}{2}-n \right) \right)!} \\ & \\  & \text{The above relationship shows that as }k\to n\text{ we see }i\to \frac{i}{2} \\ \end{align}$$

Author EGM6321.f10.team1.russo 08:10, 6 December 2010 (UTC) 70.185.113.180 10:34, 7 December 2010 (UTC)EGM6321.f10.team1.Lang.Paul

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