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Ancient Greek/Alphabet

The Greek alphabet, adopted in Attica in 403 B.C., contained 24 letters, and survives to this day, although, as is natural with most languages, the sounds of the letters have changed. While the Greeks themselves throughout the classical period wrote only in capitals, and without punctuation, this wikibook will use the current convention - that is, to use lower case letters where appropriate, to include accents and other diacritical marks, and to mark punctuation.

Pronunciation
It is difficult, but not impossible, to reconstruct the pronunciation of an ancient language. Through various methods, particularly by comparison with other languages, and observation of sounds that are confused, we can form good estimates. However, please note that scholars do not agree on the pronunciation of Ancient Greek. The following pronunciation guide is simply one reconstruction. If you ever take a formal class in Ancient Greek, you may find that your professor or teacher pronounces the words differently depending upon the reconstructed model which they learned in school.

Also, just as English pronunciation has changed over time and is different in different regions of the English speaking world, Ancient Greek pronunciation was dependent both on era and dialect. Many dialectical changes are reflected in the spelling of words (so it is not necessary to explain those here), however, it is important to note that there are differences between the pronunciation of Homeric Greek and Classical Greek.

In addition to the table above, some particular points need to be raised over pronunciation.
 * Normally, γ is pronounced with a 'hard' g sound. However, before κ, χ, γ, it is pronounced as a nasal [ŋ].
 * ι is normally a separate vowel. However, in certain circumstances, it is possible to pronounce it as a semivowel [j] (like English y). Generally, the effect is the same.

Diphthongs
Notice the short line under the long vowels α, η, ω in three instances: this is the iota subscript. It indicates that there was originally an iota after the vowel, i.e. ῳ = ωι. Some texts simply print an iota after the vowel: this is known as an iota adscript, and some texts mix the two, using a subscript with lowercase letters, and an adscript for capitals (Ωιομην for ῳομην). By Classical times, these had become monophthongs (one vowel sound) equivalent to their non-iotized counterparts, and can simply be pronounced as such. Note that when alpha takes the iota subscript, one can infer that the alpha was originally long, whereas the diphthong αι is formed from short alpha.

Aspirates
The letters θ, φ, χ are not fricatives: they are not the same as the sounds th, f, ch in thin, foot, and loch (although these are the sounds they have become in Modern Greek). These letters have hard sounds, but are pronounced with an exhalation, like an h sound. Listen hard to your voice when you pronounce the examples above: you'll find that there is a difference between the p in pot and in spot. Don't worry if you have too much difficulty making this distinction; many teachers and students prefer to use the fricative versions (th, f, kh) so that they are easier to distinguish for speakers and listeners.

Diacritical Marks
You will notice that many Greek letters have marks over them, and you will likely recognise the more common accents from other languages. In Greek, the accent does not mark stress (as in Spanish) or a change in vowel quality (as in French), but the pitch. Do not worry about remembering accents to begin with - it is quite possible to have a degree in Classics with only the vaguest knowledge of accents.

Greek has three accents: the acute or oxytone accent, written ´, and indicating a rise in pitch, the grave accent, `, indicating either a fall in pitch or a normal pitch (no one knows for sure), and a circumflex, or a perispomenon, ῀, which indicates a rise, followed by a fall. In some texts this appears as a rounded carat, like the sign ^, in others, like a tilde, ~. Accents only appear on vowels. If a diphthong is accented, the accent appears on the second character.

The diaeresis, ¨, is not strictly an accent, nor is it an umlaut, as used in German. It is mainly found on iotas to indicate that the letter should be pronounced separately from the vowel before it. For example, Verdi's opera Aïda is not pronounced 'eye-da', but 'a-yee-da'. Likewise, whereas αι is a diphthong, αϊ is two separate vowels.

More important than accents are the breathing marks. Breathings normally occur only at the beginning of a word, though they will, at times, be present in the middle - where two words have been joined together. Greek has two types of breathing, the smooth breathing, ᾿, and the rough breathing, ῾. Unlike accents, these are quite important, and you should try to learn them. A word that begins with a rough breathing on a vowel should be pronounced with an 'h'. So the Greek word ὅρος, meaning 'boundary', should be read 'horos', while ὄρος, a mountain, should be read 'oros'. Be careful never to confuse breathings with quotation marks or accents; when you write or quote Greek, the breathings should look like half-rings to clearly distinguish them from other marks. When rho occurs at the beginning of a word, it too takes a rough breathing mark and is transliterated rh, whereas when two rhos occur consecutively, the first takes a smooth breathing and the second a rough, although these breathing marks are not always indicated. Whenever upsilon occurs at the beginning of a word, it takes a rough breathing, except in the name of the letter itself. As with accent marks, breathing marks are written on the second of the two characters of a diphthong. Accents and breathings are usually written to the left of capital letters: Ἇ for ἇ.

Vowel length
In Ancient Greek, vowels may be long or short. The vowels ε and ο are always short, whereas η and ω are always long. The vowels α, ι, and υ, on the other hand, may be either short or long. It is often taught in grade school that, e.g., the "a" in "rat" is short, whereas the "a" in "rate" is long. This is not at all like vowel length in Ancient Greek. In Ancient Greek, a long vowel is literally "longer" than a short vowel, that is, its sound lasts for twice as long as the sound of a short vowel. Originally, this was the only difference between short and long vowels, but as the language evolved, the short and long vowels began to differ not only in quantity (length) but also in quality (kind).

It is not always possible to tell whether an alpha, iota, or upsilon is short or long. In dictionaries and textbooks, as here, long alphas, iotas, and upsilons are marked with a macron (¯), whereas short vowels are usually unmarked, but occasionally marked with a breve (˘). If an alpha, iota, or upsilon takes a circumflex, one may conclude that it is definitely long (see the following section), so it is not marked with a macron. If an alpha, iota, or upsilon takes an acute or a grave and it is long, there are typographic difficulties and thus it might not carry a macron even though it is supposed to (although one might occasionally see an attempt, e.g., ἀγορᾱ́, which might not render correctly in your browser).

In practice, alpha, iota, and upsilon are usually short in the word stems of Classical Greek (although the same might not be true of all Ancient Greek dialects); most long alphas in Classical Greek occur in grammatical endings, which makes it easier for the learner to remember which vowels are short and which are long. Developing a habit of pronouncing Ancient Greek words, in reading and in memorization, with full attention to vowel length, will aid the learner in remembering vowel length as well.

Accents and Morae
Ancient Greek was a language that distinguished between long and short syllables. Recall that the letters omicron and epsilon always denote short vowels for the purposes of accentuation (length can be determined differently for the sake of meter). Alpha, iota, and upsilon (α, ι, υ) can denote either short or long. but the letters eta and omega are always long, and all the diphthongs (αι, οι, ει, υι, αυ, ευ, ου) and vowels with an iota subscript (ᾳ, ῃ, ῳ) are long, except that αι and οι are usually short at the end of a word (exceptions will be noted). αι and οι are long when (1) they are not at the end of a word, (2) they end a verb in the optative mood, (3) they end noun in the locative, e.g οἴκοι (4) they end certain adverbs, e.g. αἰαῖ.

The basic unit of meter is a mora. The difference between long and short vowel is the number of morae (singular: mora) it contains. A short vowel counts as one mora, and a long vowel counts as two.

Rules for Accentuation

 * 1) Almost every Ancient Greek word has an accent, and the accent will always fall on one of the last three syllables. Examples: ποταμός (river; accent on the ultima), ἵππος (horse; accent on the penult) ἰπποπόταμος (hippopotamus; accent on the antepenult).
 * 2) The rules for accents vary based on the type of word being accented, e.g. nouns and verbs are accented differently.

Recessive Accents (Verbs)

 * 1) Most verbs have recessive accents (they go as close to the beginning of the word as possible), except infinitives and participles (which have persistent accents) and forms of a few verbs like εἰμί which have enclitic forms.
 * 2) Recessive accents always appear over the antepenult or penult. They appear over the antepenult if the last syllable is short, and over the penult if the last syllable is long.
 * 3) In the optative, αι and οι are long at the end of a word
 * 4) Contract Verbs (and other contract words) have accents that reflect the uncontracted form of the words
 * 5) * If the accented syllable is not the one that contracts, there is no effect on the accent.
 * 6) * If the accented vowel is the first of the two vowels that contract, the resulting vowel will have a circumflex
 * 7) * If the accented vowel is the second vowel in the contracted pair, it will have an acute accent.
 * 8) Consider the example of παιδευσαι: παίδευσαι 2nd person singular aorist middle imperative, (αι is short) παιδεύσαι 2nd person singular aorist active optative (αι is long), παιδεῦσαι aorist active infinitive (accent is persistent).

Persistent Accents (Nouns and Adjectives)
1. Nouns by default have persistent accent: that is to say, the accent (whichever it is) remains, as the same accent, on the same syllable (given by the principal parts of the noun) regardless of the declensional endings it might take on, unless prohibited by a rule of accentuation.

Accent Pronunciation

 * 1) When a short vowel takes the acute accent, its pitch is rising. When a long vowel takes the acute, its second mora has a rising pitch. The mora immediately following has a falling pitch, even if it is on the next word.
 * 2) A circumflex can occur only over a long vowel. It indicates that the first mora has a rising pitch and the second a falling pitch. It is only allowed to occur on the last two syllables.
 * 3) The pronunciation of the grave accent is not precisely known. An acute accent on the last syllable (the ultima) changes into a grave unless the word stands last in its clause. The grave does not occur under any other circumstances. Example: ὁρ-ῶ αὐ-τό, but αὐ-τὸ ὁ-ρῶ.
 * 4) If the last syllable is long, the accent will not fall on the third last syllable (the antepenult). (This rule has exceptions, but don't worry about them right now.)
 * 5) If the second last syllable (the penult) is long and it takes the accent:
 * The accent will be an acute if and only if the ultima is long. Examples: βαί-νω, σῴ-ζω
 * The accent will be a circumflex if and only if the ultima is short. Examples: βαῖ-νε, σῷ-ζε

Punctuation
Ancient Greek is generally punctuated in texts for the reader's convenience. Full stops and commas are used in roughly the same way as English. However, there is no exclamation mark, and the Greek question mark is used—it looks like a semicolon. Instead of colons and semicolons, Greek texts have a raised dot ( · ).

External Resources

 * Anki Deck: Ancient Greek Alphabet

title: Ancient Greek/Alphabet


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Talk:Complex Analysis

I'm aware that one already exists, but I'm taking this course and I'm writing it as I learn. Nobody will probably read this, but I'm just gonna start working on it. It's a good way to check if I understand it :-). Feel free to fix errors and stuff. This is also to learn about wikibooks. Later!

--Sciyoshi 16:19, 12 February 2006 (UTC)

 * Hey Sciyoshi,
 * What textbook are you using for your class? Nymos 17:57, 19 February 2006 (UTC)

 * Hello all, I have some expertise in complex analysis and am interested in contributing. Is anyone currently working on this book?  Does anyone have suggestions about additional material that should be added?  I was thinking that a chapter on power series would be an important part of a book on complex analysis, perhaps I will start working on that. - Brian Maurizi March 23 2010.

In section 2.1, I think that multi-valued functions should not be referred to as a "type of complex function" because complex functions were defined as maps from C to C. A multi-valued function does not meet this definition. Perhaps we could say "A related concept is the *multi-function*, which has multiple outputs for a single point in its domain. A multi-function is not a complex function by the above definition, but the concept is very useful.  For instance, the square-root "function" z^(1/2) is in fact a multi-function.  It is possible to restrict the input domain of z^(1/2) in such a way that it is a standard function, this will be discussed later." - Brian Maurizi March 23 2010

It is common in math books to have a list of notations used in the book, usually just one or two pages listing the symbols and their meaning in the book, i.e. R = real number line, $$ \Omega $$ = open set in the complex plane etc. Would this belong in "chapter 0?" - Brian Maurizi March 23 2010

 * Hi Brian, I am not working on this but I saw your questions when I was patrolling for vandalism by anonymous editors. As far as I can tell there is not much current activity on this book and it seems you are the only person to have touched it in the past 30 days. (see recent changes)
 * If you wish to do much work, you may find that creating an account helps to to keep track of your changes. (as well as making admin of the site easier for us wiki-gnomes!) Recent Runes (talk) 23:07, 23 March 2010 (UTC)

To be added
There should probably be another section on the Cauchy-Riemann integral, winding functions, zero counting, singularity detection, etc.. Fephisto (discuss • contribs) 14:51, 10 February 2015 (UTC)

 * Coming up. I intend to rework the content and the formatting of the book substantially. For any suggestions or complaints, to which I will react, please leave a message on my talk page. --Mathmensch (discuss • contribs) 15:59, 18 May 2015 (UTC)

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

title: Talk:Complex Analysis


text:
Complex Analysis/Complex Numbers

In order to fully appreciate this book, the reader should have a basic understanding of elementary mathematics (especially complex numbers and basic set theory). However, certain conventions will be used throughout the text, so an introduction is warranted.

Chapter 1: Complex Numbers

 * 1) /Introduction/
 * 2) The Topology of the Complex Plane and Stereographic Projection

title: Complex Analysis/Complex Numbers


text:
Complex Analysis/Complex Numbers/Introduction

This book assumes you have some passing familiarity with the complex numbers. Indeed much of the material in the book assumes your already familiar with the multi-variable calculus. If you have not encountered the complex numbers previously it would be a good idea to read a more detailed introduction which will have many more worked examples of arithmetic of complex numbers which this book assumes is already familiar. Such an introduction can often be found in an Algebra (or "Algebra II") text, such as the Algebra wikibook's section on complex numbers.

Intuitively a complex number z is a number written in the form:
 * $$z=x+iy$$,

where x and y are real number and i is an imaginary number that satisfies $$i^2 = -1$$. We call x the real part and y the imaginary part of z, and denote them by $$\text{Re }z$$ and $$\text{Im }z$$, respectively. Note that for the number $$z=3-2i$$, $$\text{Im }z=y=-2$$, not $$-2i$$. Also, to distinguish between complex and purely real numbers, we will often use the letters z and w for the complex numbers. It is useful to have a more formal definition of the complex numbers. For example, one frequently encounters treatments of the complex numbers that state that $$i$$ is the number so that $$i=\sqrt{-1}$$, and we then operate with $$i$$ using many of our usual rules for arithmetic. Unfortunately if one is not careful this will lead to difficulties. Not all of the usual rules for algebra carry through in the way one might expect. For example, there is a flaw in the following calculation: $$i=\sqrt{-1}=\sqrt{\frac{1}{-1}}=\frac{\sqrt{1}}{\sqrt{-1}}=\frac{1}{i}=-i$$, but is very difficult to point out the flaw without first being clear about what a complex number is, and what operations are allowed with complex numbers.

Mathematically the complex numbers are defined as an ordered pair, endowed with algebraic operations.

The most immediate consequence of this definition is that we may think of a complex number as a point lying the plane. Comparing this definition with the intuitive definition above, it is easy to see that the imaginary number i simply acts as a place holder for denoting which number belongs in the second coordinate.

We say two complex numbers are equal if and only if they are equal as ordered pairs. That is if $$z=(x,y)$$ and $$w=(u,v)$$ then z = w if and only if x = u and y = v. Put more succinctly, two complex numbers are equal iff their real parts and imaginary parts are equal.

If complex numbers were simply ordered pairs there would not really be much to say about them. But the complex numbers are ordered pairs together with several algebraic operations, and it is these operations that make the complex numbers so interesting.

Of course, we can view any real number r as being a complex number. Using our intuitive model for the complex numbers it is clear that the real number r should correspond to the complex number (r, 0), and with this identification the above operations correspond exactly to the usual definitions of addition and multiplication of real numbers. For the remainder of the text we will freely refer to a real number r as being a complex number, where the above identification is understood.

The following facts about addition and multiplication follow easily from the corresponding operators for the real numbers. Their verification is left as an exercise to the reader. Let z, w and v be complex numbers, then:

One nice feature of complex addition and multiplication is that 0 and 1 play the same role in the real numbers as they do in the complex numbers. That is 0 is the additive identity for the complex numbers (meaning z + 0 = 0 + z = z) and 1 is the multiplicative identity (meaning z · 1 = 1 · z = z).

Of course it is natural at this point to ask about subtraction and division. But stating the formula's for subtraction and division outright, we instead follow the usual course for other subjects of algebra and first discuss inverses.

Now for any two complex numbers z and w we define z − w to be z + −w. We now turn to doing the same for multiplication.

We may now of course define division as $$\tfrac{z}{w}=z\cdot\tfrac{1}{w}$$. Just as with the real numbers, division by zero remains undefined. In order for this last definition to make more sense it helps to introduce two more operations on the complex numbers. The first is the absolute value.

Of course with this definition of the absolute value, if z = (x, y) then |z| is exactly the same as the norm of the vector (x, y).

Before introducing the second definition, notice that our intuitive definition simply required us to find a number whose square was −1. Of course i2 = (−i)2 = −1, so for a starting point one could have chosen -i as the most basic imaginary number. This idea motivates the following definition.

With this definition it is an easy exercise to check that $$z\cdot\bar z=|z|^2$$, so dividing both sides by |z|2 we arrive at $$z\cdot\tfrac{\bar z}{|z|^2}=1$$. Compare this with the definition of the multiplicative inverse above.

Recall that, every point in the plane can be written using rectangular coordinates such as (x, y) where of course the numbers denote the distance from the x and y axes respectively. But the point could equally well be described using polar coordinates (r, θ), where the first number represents the distance from the origin, and the second is the angle that is made with the positive x axis when you connect the origin and the point with a line segment. Since complex numbers may be thought of simply as points in the plane, we can immediately derive a polar representation of a complex number. As usual we can let a point z = (x, y) = (r cos θ, r sin θ) where $$\textstyle r=\sqrt{x^2+y^2}$$. The choice of θ is not unique because sine and cosine are 2π periodic. A value θ for which z = (r cos θ, r sin θ) is called an argument of z. If we restrict our choice of θ so that 0 ≤ θ < 2π then the choice of θ is unique provided that z ≠ 0. This is often called the principle branch of the argument.

As a shorthand, we may write $$\operatorname{cis}\,\theta = \cos \theta + i\sin \theta$$, so $$z=r\operatorname{cis} \theta$$. This notation simplifies multiplication and taking powers, because by elementary trigonometric identities. Applying this formula can therefore simplify many calculations with complex numbers.

Using induction we can show that
 * $$z^n=r^n \operatorname{cis} (n \theta)$$,

holds for all positive integers $$n$$.

Now that we have set up the basic concept of a complex number, we continue to topological properties of the complex plane.

Exercises
 Determine $$\overline{(z_1+z_2)}$$ in terms of $$\bar z_1$$ and $$\bar z_2$$.

Determine $$\overline{z_1z_2}$$ in terms of $$\bar z_1$$ and $$\bar z_2$$.

 Show that the absolute value on the complex plane obeys the triangle inequality. That is show that:
 * $$|z_1+z_2|\leq |z_1|+|z_2|.$$

 Show that the absolute value on the complex plane obeys the reverse triangle inequality. That is show that:
 * $$|z_1+z_2|\geq \big||z_1|-|z_2|\big|.$$

 Given a non-zero complex number $$z=r\operatorname{cis}(\theta)$$ determine $$r'$$ and $$\theta'$$ so that $$\frac{1}{z}=r'\operatorname{cis}(\theta')$$.

 Determine formulas for $$\text{Re }z$$ and $$\text{Im }z$$ in terms of $$z$$ and $$\bar z$$.

 Find $$n$$ distinct complex numbers $$z_k$$, $$k = 0, \ldots, n-1$$ so that $$z_k^n=z$$. Hint: Use the formula given above for $$z_k^n$$ and the $$2\pi$$ periodicity of $$\cos(\theta)$$ and $$\sin(\theta)$$.

Next

Análise Complexa/Índice/Números Complexos/Introdução