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Talk:Electromagnetic radiation

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Electromagnetic radiation
observed electromagnetic waves from the sky as evidence by marabu birds forming circular clouds oscillation west to east then back west cited: F.O.O         Farnado Ochieng Odhiambo  — Preceding unsigned comment added by 105.161.102.41 (talk) 15:55, 17 August 2022 (UTC)

 * 1) The graphic of the em-wave is wrong (Onde_electromagnetique.svg/600px-Onde_electromagnetique.svg.png). A em-wave should look like this: https://www.miniphysics.com/wp-content/uploads/2015/09/linearly-polarized-electromagnetic-wave.png  — Preceding unsigned comment added by 78.43.236.192 (talk) 00:30, 20 November 2023 (UTC)

Electromagnetic interference
This section improperly conflates two completely different types of interference.1.72.4.68 (talk) 23:56, 18 August 2022 (UTC)
 * Which section, and care to elaborate? VQuakr (talk) 00:33, 19 August 2022 (UTC)

Wiki Education assignment: Science Communication
— Assignment last updated by AOXQueen (talk) 14:19, 17 March 2023 (UTC)

enhancing imagery and demonstration of circularly polarized light
There is an overwhelming majority of representations of light that are polar. But it would be very beneficial to have more discussion of nonpolar or circularly polarized light that give a more whole view of the concept. I would like to request someone make a representation of the circularly polarized light figure with the magnetic field also included. Psi-Archimedes (talk) 17:55, 10 March 2023 (UTC)

Momentum
Maby I got it wrong what momentum represents in the article, if not, it is impossible for anything traveling at the speed of light to have mass (a requirement of momentum), thus no electromagnetic radiation has momentum. Please explain? 2603:6011:A004:AD0C:C930:78E0:1B1E:F07D (talk) 08:08, 7 July 2023 (UTC)

 * EM radiation has no mass, but nevertheless, it carries energy and momentum. Remember, E=mc2, thus energy is equivalent to mass.  So, if there is energy in transport, there must be momentum in transport. Constant314 (talk) 12:39, 7 July 2023 (UTC)
 * Thank You, for explaining a reasoning of how EM radiation has momentum, however I have never done the math before, & am more confused now: since energy is not 0 how can mass be 0? (Seems to be your answer). I always took the E and the m as convertible potentials not necessarily current states. For example: x amount of E has has the potential to produce y amount of m in relation to c². It didn't seem right to say:
 * X amount of energy exists at the same time as Y amount of mass in relation to speed of light squared. I could be wrong? 2603:6011:A004:AD0C:C930:78E0:1B1E:F07D (talk) 06:56, 8 July 2023 (UTC)
 * Think of mass as one form that energy can assume, but energy can also assume other, massless forms, such as electromagnetic radiation. Physicists also believe that EM radiation has gravity, but it is so weak that it is not possible to test that belief.
 * When you hit a ball with a bat, the particles in the bat never touch the particles in the ball. Instead, the bat and ball exert electromagnetic forces on each other.  When momentum and energy from the bat is transferred to the ball, the momentum and energy are carried by the EM forces (light). Constant314 (talk) 23:52, 8 July 2023 (UTC)
 * Thank you for tolerating my confusion. Sorry, I only knew momentum p to equal the product of m mass & v velocity, which kept me in a box. I thought that some prankster got the momentum part added. While it would be interesting if a flashlight kicked back when switched on, i couldn't believe it. Then I came across these special formula gems, for massless p (momentum): p=E/c (sorry for doubting you), & if you want to take Planck at his observation p=hf/c, or p=h/λ 2603:6011:A004:AD0C:C930:78E0:1B1E:F07D (talk) 19:41, 9 July 2023 (UTC)
 * Yes, the flashlight does kick back, only the force is so small you cannot feel it. Photon propulsion is a possibility, but it is very inefficient in terms of power required for a given amount of thrust. Constant314 (talk) 20:15, 9 July 2023 (UTC)
 * If we could only get a star to follow our solar sail at the speed & direction that we want to go,,,,,then maybe? 2603:6011:A004:AD0C:C930:78E0:1B1E:F07D (talk) 06:16, 10 July 2023 (UTC)
 * I have to trust, but don't understand (I'm still stuck in a Newtonian box), how light reflecting off a solar sail, doesn't loose velocity at the change in direction? A q ball hitting a side rail (in a pool hall) is supposed to have v=0 for a moment and then reflect? I'm sure that there's some special relativity for light, but I just don't know what to search? 2603:6011:A004:AD0C:614A:7F80:9A4B:7EA8 (talk) 07:21, 10 July 2023 (UTC)
 * It doesn't lose velocity, but it does lose momentum and energy. The reflected light has a longer wavelength hence, less energy per photon. Constant314 (talk) 08:37, 10 July 2023 (UTC)

Electromagnetic waves and energy: tautologies abound
Regarding this sentence in the page: "EM waves carry energy, momentum and angular momentum away from their source particle and can impart those quantities to matter with which they interact."

In trying to figure out a reality for myself, being 70kg of mass, and wondering how E=MC2 relates, any description or definition of energy and how it relates to electromagnetic waves constantly seems to produce tautologies, such as what I quote from the sentence above. A definition of something can't be tautological, i.e., the saying of the same thing twice over in different words.

If energy is defined as electromagnetic radiation, then how can "electromagnetic waves carry energy" without it being a tautology? Angwhit (talk) 12:37, 20 September 2023 (UTC)

 * Easy. Energy is not defined as electromagnetic radiation. Constant314 (talk) 12:51, 20 September 2023 (UTC)

What happened?
This article used to be fairly good. Not so much anymore. Bad grammar is just a minor failure. Here is an idea: in the sections describing the various types of emr (radio, ..., gamma), actually describe them! Radical, I know. Sources, what does the type interact with (covalent bonds, nuclear forces, etc.), what are the commercial applications, and what are the observed upper and lower bounds? It should also be mentioned that while visible emr is considered to have colors (from red to violet (although some people can see into the ultraviolet) it is not true that an object of a certain color necessarily emits photons of that frequency/wave length. (i.e photons of a certain color are one of several ways to evoke that color.) Also, it is not true that it is energy that distinguishes x-rays from gamma rays - at least by some common definitions. (It is the type of source so a higher energy photon may be considered an x-ray compared to a lower energy gamma ray.) I'm a chemist and I think it would be quite useful to describe the types of energy transitions that absorb/emit the various types of emr. (for instance, gamma rays interact with the nucleus of atoms and not (at least, the interaction is rare) with the electrons. (The article goes on and on about ionizing radiation. It's quite confused. I guess it's possible that a nuclear reaction is considered "ionizing" if the number of protons changes, but as far as I know "ionizing radiation" is radiation that knocks an electron off of an atom. Full stop. I'd have to check to see what the energy is of the 1s electrons in U or Pu, but I don't think gamma rays interact with the electron shells. (I may be wrong there.) One question that should be answered for each type of radiation is their sources. The article on Gamma Ray Astronomy claims that in 2018 the highest energy gamma ray photon ever detected was seen (1.4 peta eV) which is way off most of the charts in this article. And that's another thing that should probably be mentioned - as the subject moves from radio waves to light waves to gamma rays, their units of measure often change from wave length to wave number to frequency to electron volts. While all of these are interconvertable, it's probably notable enough to deserve mention. So, I wonder what the longest wavelength photon that's ever been observed is? Will I find it here? Why not?71.30.94.234 (talk) 10:43, 2 November 2023 (UTC)

 * I agree with most of these suggestions. As you say, the "Electromagnetic spectrum" section should have simpler descriptions of how each type of radiation is generated and detected, as well as the instruments used to detect it, what it is used for, and the units used to measure it. It also needs sources, most of it is unsourced.
 * On the other hand I disagree with you about ionizing and nonionizing radiation: the article should emphasize this difference. This is the main measure of the health risks of radiation and is widely misunderstood by the public. There is no debate that vacuum UV, x-rays, and gamma rays ionize matter. That was how they were first detected, and is the basis of most radiation detectors. As a chemist, I'm surprised you don't know that. --ChetvornoTALK 19:09, 2 November 2023 (UTC)

Electromagnetic Radiation vs Electromagnetic Field
Somehow Wikipedia does not see a difference between Electromagnetic Field which stays with the charge and electromagnetic radiation which leaves the charge. Here we should say in the definition that there are "many non radiant EM fields causing a great many physical things like an electric motor". On line the confusion has some saying motors and wires always radiate photons. Bill field pulse (talk) 21:20, 21 January 2024 (UTC)

 * That is mostly incorrect. This is stuff is described with sloppy language all over the place and we are supposed to guess from context what is really meant.  As a result people get misconceptions about what is meant be the electromagnetic (EM) field.  For some physicists (Feynman) the EM field is nothing, but numbers attached to points in space.  For others the EM field is physical, but still described completely by numbers attached to points in space.  The numbers obey Maxwell's and the other equations of electromagnetic field theory.  The field does not move.  There is only one EM field, and it fills all of space.  Since there is only one, it is always singular.  It should always be the EM field and never a EM field or an EM field. Charged particles do not have EM fields.  Rather, they influence the value of the EM field around them.  Far away particles can also influence the local field by EM radiation.
 * When you hold your hand out in bright sunshine, you can feel EM radiation. EM radiation is a real physical phenomenon.  The EM field is a tool invented by humans.  Of course, the EM field and EM radiation are not the same.  However, the concept of the EM field readily accommodates both EM radiation and static EM forces.  There is no great wrong here that needs to be righted. Constant314 (talk) 02:38, 23 January 2024 (UTC)

 * I think maybe Bill field pulse is referring to the difference between the near fields and far fields of a source of electromagnetic radiation. Bill, the article does mention this issue: the introduction says "Electromagnetic radiation is associated with those EM waves that are free to propagate themselves ("radiate") without the continuing influence of the moving charges that produced them . . ." although maybe this could be stated more clearly.

 * Electromagnetic waves are defined as the part of the field of accelerating charges that can carry energy arbitrarily far away from the charges. Unless they are reflected back to the source charges, they cause the source to lose (radiate) energy.  Close to moving charges there are other components of the oscillating electric and magnetic field which just store energy in the space around the charges.  If there is a conductor near the moving charges, the alternating electric field can induce a charge in it by electrostatic induction, and the alternating magnetic field can induce current in it by electromagnetic induction.  The latter is how motors and generators work.  But if there are no nearby conductors, the energy just stays close to the charges, stored alternately in the electric and magnetic field.  These are usually called the induction fields or near fields.  --ChetvornoTALK 11:41, 23 January 2024 (UTC)
 * Chetvorno, I appreciate your effort to see intelligence in my comments. Classically experimenters thought a field existed around any charge and the field obeyed coulomb law getting weaker with distance. When hairs repel due to static electricity it is said that the surplus electrons on each hair are repelling. The elections stay as far apart as possible. When current travels on a wire the free electrons are said to be on the outside of the wire. This is all considered evidence that electrons have fields on them which reach distances far beyond simple atomic distances.  If a charge moves quickly the rather large field around it would need to be super rigid not to lag behind. I believe it is not infinitely rigid and that it arrives a very tiny instant later (at the speed of light is very very rigid but not infinitely rigid.) This slight non infinite rigidity is necessary for a field around a fast charge to differ from a field around a slow charge. The slightly compressed field causes magnetism. Bill field pulse (talk) 21:27, 23 January 2024 (UTC)
 * John J Barton mentioned the near and far field again and I had a second look at your answer, and the article. You seem to be right the far field and near field seem to capture the difference I am talking about without needing to get into particle effects. That the near field effect diminishes with 1/r x 1/r makes perfect sense since it is governed by Coulombs law. That the far field which is the essentially the EM radiation requiring photons diminishes with 1/r also is possible because the photons don't loose strength they only become further apart.
 * I think the EM Field article only refers to far field effects and even has a drawing of this. I would like to see near and far field effect and link holding a higher place in the EM Field article. Having only the picture of the far field effect misses the near field reality. What do you think?
 * Thanks for something I did not give nearly adequate thought the first time I read your reply. Sorry for that . Bill field pulse (talk) 21:18, 27 January 2024 (UTC)
 * Thank you for taking a little time to discuss this it is much appreciated. When you have an electron surplus on two metal coated balloons separated a small distance and no radiation. How do you see them repelling each other what goes between the balloons. I am curious to understand the mechanism you see. Perhaps I am only using different terms and need to learn yours?   Bill field pulse (talk) 21:10, 23 January 2024 (UTC)
 * I am not clear on why you think Wikipedia "does not see a difference"; we do have a separate article at Electromagnetic field. VQuakr (talk) 22:24, 23 January 2024 (UTC)
 * The electromagnetic field is something which is around a charge and drops of with distance squared. The primary cause of quantization of the electromagnetic field is that charges are quantized. The net steady EM field we measure is the result of "billions" of tiny EM Field caused by "billions" of moving charges. Electromagnetic radiation is very different. An electron continuously produces an EM field in all directions and in rare instances it sheds or absorbs a whole photon as radiation.
 * Look at the article electromagnetic field there is a picture of an entity which moves in one direction without loosing strength; it is a classic diagram of electromagnetic radiation. Look at the electromagnetic radiation article, it is the same diagram!!! Why not say "see electromagnetic radiation in the electromagnetic field article".
 * It is very easy for beginners to mix up the em field moving out in all radial directions around a charge at the speed of light and em radiation moving out at the speed of light in a single direction; both must move at the speed of light and I can explain why in detail if you are interested. Bill field pulse (talk) 21:39, 24 January 2024 (UTC)
 * I think you have some things confused. The drawing Onde electromagnetique.svg on both pages is showing the same thing: an electromagnetic wave, as it says in the caption of both.  This drawing is in the Electromagnetic field article because electromagnetic radiation is one phenomenon or aspect of electromagnetic fields.  I don't see anything misleading or confusing in these two articles.  What specific changes or edits would you like to make? --ChetvornoTALK 01:49, 25 January 2024 (UTC)
 * I would like to see the Electromagnetic radiation article focus primarily on electromagnetic radiation. I would like to see the electromagnetic field article focus primarily upon non radiant fields with a note linking to the electromagnetic radiation article. Lastly, I would like to see quantization of the electromagnetic field focus on the quantization of charges whereby all the small fields around charges sum together to produce a net EM field. Ideally, the quantization of electromagnetic radiation would be in a section with a heading "The quantization of Electromagnetic Radiation. Bill field pulse (talk) 18:22, 25 January 2024 (UTC)
 * Convenience links: Quantization of the electromagnetic field, Electromagnetic field. I don't feel your comments so far are specific enough to be actionable. Can you present what you are proposing (or a part of it) in a clear "change x to y" format with the exact text and sources you would like to add/change? VQuakr (talk) 19:14, 25 January 2024 (UTC)
 * I made a very small change to start with..... I added "if any" to radiation out of a wire and removed "propagated at" since radiation has a frequency or is propagated with a frequency.
 * You can delete or reword. Note the magnetic field around a wire is not radiation. Radiation occurs when it is intended or when the design is poor. Bill field pulse (talk) 20:04, 25 January 2024 (UTC)
 * Reverted. Radiation occurs but can be minimized by the nature of the design. VQuakr (talk) 20:24, 25 January 2024 (UTC)
 * Yes there are very few photons produced inside household wires. It is 99.999% the EM Field that things like current meters measure Bill field pulse (talk) 18:35, 26 January 2024 (UTC)
 * Any time there is any change in the electromagnetic field, it is communicated to the rest of the universe as radiation. Constant314 (talk) 18:44, 26 January 2024 (UTC)
 * I think that only changes due to electrons changing energy levels are communicated via photons. The continuous changes in ac current are communicated by the field itself which drops quite fast with distance squared. Bill field pulse (talk) 18:54, 26 January 2024 (UTC)
 * It is still radiation, regardless of the mechanism. Constant314 (talk) 19:39, 26 January 2024 (UTC)
 * I think I visualize a mechanism and you don't see the point. but my silly mechanism lets me look at how quarks might move and how gravity and strong force might arise perhaps leading to experiments Maxwell used mechanisms Einstein used an elevator and stuff to visualize things. Bill field pulse (talk) 20:09, 26 January 2024 (UTC)
 * Even Newton said he does not know the mechanism for gravity but if he knew of quarks you can bet he would have come up with a mechanism Bill field pulse (talk) 20:11, 26 January 2024 (UTC)
 * I think I visualize a mechanism... is this your theory? This is an encyclopedia, not a journal: we don't publish original work. VQuakr (talk) 20:44, 26 January 2024 (UTC)
 * I shall try not to do any thinking which is original Bill field pulse (talk) 19:38, 27 January 2024 (UTC)

Electro magnetic waves
Electro magnetic wages class 12th 2409:40F0:102F:E17F:D0DB:D2FF:FECC:A872 (talk) 06:23, 21 April 2024 (UTC)

title: Talk:Electromagnetic radiation


text:
Endosymbiont

An endosymbiont or endobiont is an organism that lives within the body or cells of another organism. Typically the two organisms are in a mutualistic relationship. Examples are nitrogen-fixing bacteria (called rhizobia), which live in the root nodules of legumes, single-cell algae inside reef-building corals and bacterial endosymbionts that provide essential nutrients to insects.

Endosymbiosis played key roles in the development of eukaryotes and plants. Roughly 2.2 billion years ago an archaea absorbed a bacterium through phagocytosis that eventually became the mitochondria that provide energy to all living cells. Approximately 1 billion years ago, other cells absorbed cyanobacteria that eventually became chloroplasts, organelles that produce energy from sunlight. Some 100 million years ago, UCYN-A, a nitrogen-fixing bacteria became an endosymbiont of marine algae Braarudosphaera bigelowii, eventually evolving into a nitroplast. Similarly, Diatoms in the family Rhopalodiaceae have evolved a diazoplast, a nitrogen-fixing organelle.

Symbionts are either obligate (require their host to survive) or facultative (can survive independently). The most common examples of obligate endosymbiosis are mitochondria and chloroplasts, which reproduce via mitosis in tandem with their host cells. Some human parasites, e.g. Wuchereria bancrofti and Mansonella perstans, thrive in their intermediate insect hosts because of an obligate endosymbiosis with Wolbachia spp. They can both be eliminated by treatments that target their bacterial host.

Etymology
Endosymbiosis comes from the Greek: ἔνδον endon "within", σύν syn "together" and βίωσις biosis "living".

Symbiogenesis
Symbiogenesis theory holds that eukaryotes evolved via absorbing prokaryotes. Typically, one organism envelopes a bacterium and the two evolve a mutualistic relationship. The absorbed bacteria (the endosymbiont) eventually lives exclusively within the host cells. This fits the concept of observed organelle development.

Typically the endosymbiont's genome shrinks, discarding genes whose roles are displaced by the host. For example the Hodgkinia genome of Magicicada cicadas is much different from the prior freestanding bacteria. The cicada life cycle involves years of stasis underground. The symbiont produces many generations during this phase, experiencing little selection pressure, allowing their genomes to diversify. Selection is episodic (when the cicadas reproduce). The original Hodgkinia genome split into three much simpler endosymbionts, each encoding only a few genes—an instance of punctuated equilibrium producing distinct lineages. The host requires all three symbionts.

Transmission
Symbiont transmission is the process where the host acquires its symbiont. Since symbionts are not produced by host cells, they must find their own way to reproduce and populate daughter cells as host cells divide. Horizontal, vertical, and mixed-mode (hybrid of horizonal and vertical) transmission are the three paths for symbiont transfer.

Horizontal
Horizontal symbiont transfer (horizontal transmission) is a process where a host acquires a facultative symbiont from the environment or another host. The Rhizobia-Legume symbiosis (bacteria-plant endosymbiosis) is a prime example of this modality. The Rhizobia-legume symbiotic relationship is important for processes such as the formation of root nodules. It starts with flavonoids released by the legume host, which causes the rhizobia species (endosymbiont) to activate its Nod genes. These Nod genes generate lipooligosaccharide signals that the legume detects, leading to root nodule formation. This process bleeds into other processes such as nitrogen fixation in plants. The evolutionary advantage of such an interaction allows genetic exchange between both organisms involved to increase the propensity for novel functions as seen in the plant-bacterium interaction (holobiont formation).

Vertical
Vertical transmission takes place when the symbiont moves directly from parent to offspring. In horizontal transmission each generation acquires symbionts from the environment. An example is nitrogen-fixing bacteria in certain plant roots, such as pea aphid symbionts. A third type is mixed-mode transmission, where symbionts move horizontally for some generations, after which they are acquired vertically.

Wigglesworthia, a tsetse fly symbiont, is vertically transmitted (via mother's milk). When a symbiont reaches this stage, it resembles a cellular organelle, similar to mitochondria or chloroplasts. In vertical transmission, the symbionts do not need to survive independently, often leading them to have a reduced genome. For instance, pea aphid symbionts have lost genes for essential molecules and rely on the host to supply them. In return, the symbionts synthesize essential amino acids for the aphid host. When a symbiont reaches this stage, it begins to resemble a cellular organelle, similar to mitochondria or chloroplasts. Such dependent hosts and symbionts form a holobiont. In the event of a bottleneck, a decrease in symbiont diversity could compromise host-symbiont interactions, as deleterious mutations accumulate.

Invertebrates
The best-studied examples of endosymbiosis are in invertebrates. These symbioses affect organisms with global impact, including Symbiodinium (corals), or Wolbachia (insects). Many insect agricultural pests and human disease vectors have intimate relationships with primary endosymbionts.

Insects

Scientists classify insect endosymbionts as Primary or Secondary. Primary endosymbionts (P-endosymbionts) have been associated with their insect hosts for millions of years (from ten to several hundred million years). They form obligate associations and display cospeciation with their insect hosts. Secondary endosymbionts more recently associated with their hosts, may be horizontally transferred, live in the hemolymph of the insects (not specialized bacteriocytes, see below), and are not obligate.

Primary
Among primary endosymbionts of insects, the best-studied are the pea aphid (Acyrthosiphon pisum) and its endosymbiont Buchnera sp. APS, the tsetse fly Glossina morsitans morsitans and its endosymbiont Wigglesworthia glossinidia brevipalpis and the endosymbiotic protists in lower termites. As with endosymbiosis in other insects, the symbiosis is obligate. Nutritionally-enhanced diets allow symbiont-free specimens to survive, but they are unhealthy, and at best survive only a few generations.

In some insect groups, these endosymbionts live in specialized insect cells called bacteriocytes (also called mycetocytes), and are maternally-transmitted, i.e. the mother transmits her endosymbionts to her offspring. In some cases, the bacteria are transmitted in the egg, as in Buchnera; in others like Wigglesworthia, they are transmitted via milk to the embryo. In termites, the endosymbionts reside within the hindguts and are transmitted through trophallaxis among colony members.

Primary endosymbionts are thought to help the host either by providing essential nutrients or by metabolizing insect waste products into safer forms. For example, the putative primary role of Buchnera is to synthesize essential amino acids that the aphid cannot acquire from its diet of plant sap. The primary role of Wigglesworthia is to synthesize vitamins that the tsetse fly does not get from the blood that it eats. In lower termites, the endosymbiotic protists play a major role in the digestion of lignocellulosic materials that constitute a bulk of the termites' diet.

Bacteria benefit from the reduced exposure to predators and competition from other bacterial species, the ample supply of nutrients and relative environmental stability inside the host.

Primary endosymbionts of insects have among the smallest of known bacterial genomes and have lost many genes commonly found in closely related bacteria. One theory claimed that some of these genes are not needed in the environment of the host insect cell. A complementary theory suggests that the relatively small numbers of bacteria inside each insect decrease the efficiency of natural selection in 'purging' deleterious mutations and small mutations from the population, resulting in a loss of genes over many millions of years. Research in which a parallel phylogeny of bacteria and insects was inferred supports the belief that primary endosymbionts are transferred only vertically.

Attacking obligate bacterial endosymbionts may present a way to control their hosts, many of which are pests or human disease carriers. For example, aphids are crop pests and the tsetse fly carries the organism Trypanosoma brucei that causes African sleeping sickness. Studying insect endosymbionts can aid understanding the origins of symbioses in general, as a proxy for understanding endosymbiosis in other species.

The best-studied ant endosymbionts are Blochmannia bacteria, which are the primary endosymbiont of Camponotus ants. In 2018 a new ant-associated symbiont, Candidatus Westeberhardia Cardiocondylae, was discovered in Cardiocondyla. It is reported to be a primary symbiont.

Secondary

The pea aphid (Acyrthosiphon pisum) contains at least three secondary endosymbionts, Hamiltonella defensa, Regiella insecticola, and Serratia symbiotica. Hamiltonella defensa defends its aphid host from parasitoid wasps. This symbiosis replaces lost elements of the insect's immune response.

One of the best-understood defensive symbionts is the spiral bacteria Spiroplasma poulsonii. Spiroplasma sp. can be reproductive manipulators, but also defensive symbionts of Drosophila flies. In Drosophila neotestacea, S. poulsonii has spread across North America owing to its ability to defend its fly host against nematode parasites. This defence is mediated by toxins called "ribosome-inactivating proteins" that attack the molecular machinery of invading parasites. These toxins represent one of the first understood examples of a defensive symbiosis with a mechanistic understanding for defensive symbiosis between an insect endosymbiont and its host.

Sodalis glossinidius is a secondary endosymbiont of tsetse flies that lives inter- and intracellularly in various host tissues, including the midgut and hemolymph. Phylogenetic studies do not report a correlation between evolution of Sodalis and tsetse. Unlike Wigglesworthia, Sodalis has been cultured in vitro.

Cardinium and many other insects have secondary endosymbionts.

Marine
Extracellular endosymbionts are represented in all four extant classes of Echinodermata (Crinoidea, Ophiuroidea, Echinoidea, and Holothuroidea). Little is known of the nature of the association (mode of infection, transmission, metabolic requirements, etc.) but phylogenetic analysis indicates that these symbionts belong to the class Alphaproteobacteria, relating them to Rhizobium and Thiobacillus. Other studies indicate that these subcuticular bacteria may be both abundant within their hosts and widely distributed among the Echinoderms.

Some marine oligochaeta (e.g., Olavius algarvensis and Inanidrillus spp.) have obligate extracellular endosymbionts that fill the entire body of their host. These marine worms are nutritionally dependent on their symbiotic chemoautotrophic bacteria lacking any digestive or excretory system (no gut, mouth, or nephridia).

The sea slug Elysia chlorotica's endosymbiont is the algae Vaucheria litorea. The jellyfish Mastigias have a similar relationship with an algae. Elysia chlorotica forms this relationship intracellularly with the algae's chloroplasts. These chloroplasts retain their photosynthetic capabilities and structures for several months after entering the slug's cells.

Trichoplax have two bacterial endosymbionts. Ruthmannia lives inside the animal's digestive cells. Grellia lives permanently inside the endoplasmic reticulum (ER), the first known symbiont to do so.

Paracatenula is a flatworm which have lived in symbiosis with an endosymbiotic bacteria for 500 million years. The bacteria produce numerous small, droplet-like vesicles that provide the host with needed nutrients.

Dinoflagellates
Dinoflagellate endosymbionts of the genus Symbiodinium, commonly known as zooxanthellae, are found in corals, mollusks (esp. giant clams, the Tridacna), sponges, and the unicellular foraminifera. These endosymbionts capture sunlight and provide their hosts with energy via carbonate deposition.

Previously thought to be a single species, molecular phylogenetic evidence reported diversity in Symbiodinium. In some cases, the host requires a specific Symbiodinium clade. More often, however, the distribution is ecological, with symbionts switching among hosts with ease. When reefs become environmentally stressed, this distribution is related to the observed pattern of coral bleaching and recovery. Thus, the distribution of Symbiodinium on coral reefs and its role in coral bleaching is an important in coral reef ecology.

Phytoplankton
In marine environments,  endosymbiont relationships are especially prevalent in oligotrophic or nutrient-poor regions of the ocean like that of the North Atlantic. In such waters, cell growth of larger phytoplankton such as diatoms is limited by (insufficient) nitrate concentrations. Endosymbiotic bacteria fix nitrogen for their hosts and in turn receive organic carbon from photosynthesis. These symbioses play an important role in global carbon cycling.

One known symbiosis between the diatom Hemialus spp. and the cyanobacterium Richelia intracellularis has been reported in North Atlantic, Mediterranean, and Pacific waters. Richelia is found within the diatom frustule of Hemiaulus spp., and has a reduced genome. A 2011 study measured nitrogen fixation by the cyanobacterial host Richelia intracellularis well above intracellular requirements, and found the cyanobacterium was likely fixing nitrogen for its host. Additionally, both host and symbiont cell growth were much greater than free-living Richelia intracellularis or symbiont-free Hemiaulus spp. The Hemaiulus-Richelia symbiosis is not obligatory, especially in nitrogen-replete areas.

Richelia intracellularis is also found in Rhizosolenia spp., a diatom found in oligotrophic oceans. Compared to the Hemaiulus host, the endosymbiosis with Rhizosolenia is much more consistent, and Richelia intracellularis is generally found in Rhizosolenia. There are some asymbiotic (occurs without an endosymbiont) Rhizosolenia, however there appears to be mechanisms limiting growth of these organisms in low nutrient conditions. Cell division for both the diatom host and cyanobacterial symbiont can be uncoupled and mechanisms for passing bacterial symbionts to daughter cells during cell division are still relatively unknown.

Other endosymbiosis with nitrogen fixers in open oceans include Calothrix in Chaetoceros spp. and UNCY-A in prymnesiophyte microalga. The Chaetoceros-Calothrix endosymbiosis is hypothesized to be more recent, as the Calothrix genome is generally intact. While other species like that of the UNCY-A symbiont and Richelia have reduced genomes. This reduction in genome size occurs within nitrogen metabolism pathways indicating endosymbiont species are generating nitrogen for their hosts and losing the ability to use this nitrogen independently. This endosymbiont reduction in genome size, might be a step that occurred in the evolution of organelles (above).

Protists
Mixotricha paradoxa is a protozoan that lacks mitochondria. However, spherical bacteria live inside the cell and serve the function of the mitochondria. Mixotricha has three other species of symbionts that live on the surface of the cell.

Paramecium bursaria, a species of ciliate, has a mutualistic symbiotic relationship with green alga called Zoochlorella. The algae live in its cytoplasm.

Platyophrya chlorelligera is a freshwater ciliate that harbors Chlorella that perform photosynthesis.

Strombidium purpureum is a marine ciliate that uses endosymbiotic, purple, non-sulphur bacteria for anoxygenic photosynthesis.

Paulinella chromatophora is a freshwater amoeboid that has a cyanobacterium endosymbiont.

Many foraminifera are hosts to several types of algae, such as red algae, diatoms, dinoflagellates and chlorophyta. These endosymbionts can be transmitted vertically to the next generation via asexual reproduction of the host, but because the endosymbionts are larger than the foraminiferal gametes, they need to acquire algae horizontally following sexual reproduction.

Several species of radiolaria have photosynthetic symbionts. In some species the host digests algae to keep the population at a constant level.

Hatena arenicola is a flagellate protist with a complicated feeding apparatus that feeds on other microbes. When it engulfs a green Nephroselmis alga, the feeding apparatus disappears and it becomes photosynthetic. During mitosis the algae is transferred to only one of the daughter cells, while the other cell restarts the cycle.

In 1966, biologist Kwang W. Jeon found that a lab strain of Amoeba proteus had been infected by bacteria that lived inside the cytoplasmic vacuoles. This infection killed almost all of the infected protists. After the equivalent of 40 host generations, the two organisms become mutually interdependent. A genetic exchange between the prokaryotes and protists occurred.

Vertebrates
The spotted salamander (Ambystoma maculatum) lives in a relationship with the algae Oophila amblystomatis, which grows in its egg cases.

Plants
All vascular plants harbor endosymbionts or endophytes in this context. They include bacteria, fungi, viruses, protozoa and even microalgae. Endophytes aid in processes such as growth and development, nutrient uptake, and defense against biotic and abiotic stresses like drought, salinity, heat, and herbivores.

Plant symbionts can be categorized into epiphytic, endophytic, and mycorrhizal. These relations can also be categorized as beneficial, mutualistic, neutral, and pathogenic. Microorganisms living as endosymbionts in plants can enhance their host's primary productivity either by producing or capturing important resources. These endosymbionts can also enhance plant productivity by producing toxic metabolites that aid plant defenses against herbivores.

Plants are dependent on plastid or chloroplast organelles. The chloroplast is derived from a cyanobacterial primary endosymbiosis that began over one billion years ago. An oxygenic, photosynthetic free-living cyanobacterium was engulfed and kept by a heterotrophic protist and eventually evolved into the present intracellular organelle.

Mycorrhizal endosymbionts appear only in fungi.

Typically, plant endosymbiosis studies focus on a single category or species to better understand their individual biological processes and functions.

Fungal endophytes
Fungal endophytes can be found in all plant tissues. Fungi living below the ground amidst plant roots are known as mycorrhiza, but are further categorized based on their location inside the root, with prefixes such as ecto, endo, arbuscular, ericoid, etc. Fungal endosymbionts that live in the roots and extend their extraradical hyphae into the outer rhizosphere are known as ectendosymbionts.

Arbuscular Mycorrhizal Fungi (AMF)
Arbuscular mycorrhizal fungi or AMF are the most diverse plant microbial endosymbionts. With exceptions such as the Ericaceae family, almost all vascular plants harbor AMF endosymbionts as endo and ecto as well. AMF plant endosymbionts systematically colonize plant roots and help the plant host acquire soil nutrients such as nitrogen. In return it absorbs plant organic carbon products. Plant root exudates contain diverse secondary metabolites, especially flavonoids and strigolactones that act as chemical signals and attracts the AMF. AMF Gigaspora margarita lives as a plant endosymbiont and also harbors further endosymbiont intracytoplasmic bacterium-like organisms. AMF generally promote plant health and growth and alleviate abiotic stresses such as salinity, drought, heat, poor nutrition, and metal toxicity. Individual AMF species have different effects in different hosts – introducing the AMF of one plant to another plant can reduce the latter's growth.

Endophytic fungi
Endophytic fungi in mutualistic relations directly benefit and benefit from their host plants. They also can help their hosts succeed in polluted environments such as those contaminated with toxic metals. Fungal endophytes are taxonomically diverse and are divided into categories based on mode of transmission, biodiversity, in planta colonization and host plant type. Clavicipitaceous fungi systematically colonize temperate season grasses. Non-clavicipitaceous fungi colonize higher plants and even roots and divide into subcategories. Aureobasidium and preussia species of endophytic fungi isolated from Boswellia sacra produce indole acetic acid hormone to promote plant health and development.

Aphids can be found in most plants. Carnivorous ladybirds are aphid predators and are used in pest control. Plant endophytic fungus Neotyphodium lolii produces alkaloid mycotoxins in response to aphid invasions. In response, ladybird predators exhibited reduced fertility and abnormal reproduction, suggesting that the mycotoxins are transmitted along the food chain and affect the predators.

Endophytic bacteria
Endophytic bacteria belong to a diverse group of plant endosymbionts characterized by systematic colonization of plant tissues. The most common genera include Pseudomonas, Bacillus, Acinetobacter, Actinobacteria, Sphingomonas. Some endophytic bacteria, such as Bacillus amyloliquefaciens, a seed-born endophytic bacteria, produce plant growth by producing gibberellins, which are potent plant growth hormones. Bacillus amyloliquefaciens promotes the taller height of transgenic dwarf rice plants. Some endophytic bacteria genera additionally belong to the Enterobacteriaceae family. Endophytic bacteria typically colonize the leaf tissues from plant roots, but can also enter the plant through the leaves through leaf stomata. Generally, the endophytic bacteria are isolated from the plant tissues by surface sterilization of the plant tissue in a sterile environment. Passenger endophytic bacteria eventually colonize inner tissue of plant by stochastic events while True endophytes possess adaptive traits because of which they live strictly in association with plants. The in vitro-cultivated endophytic bacteria association with plants is considered a more intimate relationship that helps plants acclimatize to conditions and promotes health and growth. Endophytic bacteria are considered to be plant's essential endosymbionts because virtually all plants harbor them, and these endosymbionts play essential roles in host survival. This endosymbiotic relation is important in terms of ecology, evolution and diversity. Endophytic bacteria such as Sphingomonas sp. and Serratia sp. that are isolated from arid land plants regulate endogenous hormone content and promote growth.

Archaea endosymbionts
Archaea are members of most microbiomes. While archaea are abundant in extreme environments, they are less abundant and diverse in association with eukaryotic hosts. Nevertheless, archaea are a substantial constituent of plant-associated ecosystems in the aboveground and belowground phytobiome, and play a role in host plant’s health, growth and survival amid biotic and abiotic stresses. However, few studies have investigated the role of archaea in plant health and its symbiotic relationships. Most plant endosymbiosis studies focus on fungal or bacteria using metagenomic approaches.

The characterization of archaea includes crop plants such as rice and maize, but also aquatic plants. The abundance of archaea varies by tissue type; for example archaea are more abundant in the rhizosphere than the phyllosphere and endosphere. This archaeal abundance is associated with plant species type, environment and the plant’s developmental stage. In a study on plant genotype-specific archaeal and bacterial endophytes, 35% of archaeal sequences were detected in overall sequences (achieved using amplicon sequencing and verified by real time-PCR). The archaeal sequences belong to the phyla Thaumarchaeota, Crenarchaeota, and Euryarchaeota.

Bacteria
Some Betaproteobacteria have Gammaproteobacteria endosymbionts.

Fungi
Fungi host endohyphal bacteria; the effects of the bacteria are not well studied. Many such fungi in turn live within plants. These fungi are otherwise known as fungal endophytes. It is hypothesized that the fungi offers a safe haven for the bacteria, and the diverse bacteria that they attract create a micro-ecosystem.

These interactions may impact the way that fungi interact with the environment by modulating their phenotypes. The bacteria do this by altering the fungi's gene expression. For example, Luteibacter sp. has been shown to naturally infect the ascomycetous endophyte Pestalotiopsis sp. isolated from ''Platycladus orientalis.  The Luteibacter'' sp. influences the auxin and enzyme production within its host, which, in turn, may influence the effect the fungus has on its plant host''.  Another interesting example of a bacteria living in symbiosis with a fungus is the fungus Mortierella. This soil-dwelling fungus lives in close association with a toxin-producing bacteria, Mycoavidus'', which helps the fungus defend against nematodes.

Virus endosymbionts
The human genome project found several thousand endogenous retroviruses, endogenous viral elements in the genome that closely resemble and can be derived from retroviruses, organized into 24 families.

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

title: Endosymbiont


text:
Exponential function

The exponential function is a mathematical function denoted by $$f(x)=\exp(x)$$ or $$e^x$$ (where the argument $x$ is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the operation of taking powers of a number (repeated multiplication), but various modern definitions allow it to be rigorously extended to all real arguments $$x$$, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics".

The functions $$f(x) = b^x$$ for positive real numbers $$b$$ are also known as exponential functions, and satisfy the exponentiation identity:$$b^{x+y} = b^x b^y \text{ for all } x,y\in\mathbb{R}.$$This implies $$b^n= b\times\cdots\times b$$ (with $$n$$ factors) for positive integers $$n$$, where $$b=b^1$$, relating exponential functions to the elementary notion of exponentiation. The natural base $$e = \exp(1)=2.71828\ldots$$ is a ubiquitous mathematical constant called Euler's number. To distinguish it, $$\exp(x)=e^x$$ is called the exponential function or the natural exponential function: it is the unique real-valued function of a real variable whose derivative is itself and whose value at $f(x) = ex$ is $f(x) = xr$: "$\exp'(x)=\exp(x)$ for all $x\in \R$, and $\exp(0)=1.$"The relation $$b^x = e^{x\ln b}$$ for $$b>0$$ and real or complex $$x$$ allows general exponential functions to be expressed in terms of the natural exponential.

More generally, especially in applied settings, any function $$f:\mathbb{R}\to\mathbb{R}$$ defined by

$$f(x)=c e^{ax}=c b^{kx}, \text{ with }k=a/ \ln b,\ a\neq 0,\ b ,c>0$$

is also known as an exponential function, as it solves the initial value problem $$f'=af,\ f(0)=c$$, meaning its rate of change at each point is proportional to the value of the function at that point. This behavior models diverse phenomena in the biological, physical, and social sciences, for example the unconstrained growth of a self-reproducing population, the decay of a radioactive element, the compound interest accruing on a financial fund, or a growing body of manufacturing expertise.

The exponential function can also be defined as a power series, which is readily applied to real numbers, complex numbers, and even matrices. The complex exponential function $$\exp:\mathbb{C}\to\mathbb{C}$$ takes on all complex values except for 0 and is closely related to the complex trigonometric functions by Euler's formula: "$e^{x+iy} = e^x\cos(y) \,+\, i \,e^x\sin(y).$"Motivated by its more abstract properties and characterizations, the exponential function can be generalized to much larger contexts such as square matrices and Lie groups. Even further, the differential equation definition can be generalized to a Riemannian manifold.

The exponential function for real numbers is a bijection from $$\mathbb{R}$$ to the interval $$(0,\infty)$$. Its inverse function is the natural logarithm, denoted $\ln$, $\log$, or $\log_e$. Some old texts refer to the exponential function as the antilogarithm.

Graph
The graph of $$y=e^x$$ is upward-sloping, and increases faster as $x$ increases. The graph always lies above the $x$-axis, but becomes arbitrarily close to it for large negative $x$; thus, the $x$-axis is a horizontal asymptote. The equation $$\tfrac{d}{dx}e^x = e^x$$ means that the slope of the tangent to the graph at each point is equal to its $y$-coordinate at that point.

Relation to more general exponential functions
The exponential function $$f(x) = e^x$$ is sometimes called the natural exponential function in order to distinguish it from the other exponential functions. The study of any exponential function can easily be reduced to that of the natural exponential function, since per definition, for positive $b$, $$ b^x \mathrel{\stackrel{\text{def}}{=}} e^{x\ln b} $$

As functions of a real variable, exponential functions are uniquely characterized by the fact that the derivative of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base $b$: $$\frac{d}{dx} b^x = \frac{d}{dx} e^{x\ln (b)} = e^{x\ln (b)} \ln (b) = b^x \ln (b).$$

For $f(x,y) = xy$, the function $$b^x$$ is increasing (as depicted for $−Wn(−1)$ and $0$), because $$\ln b>0$$ makes the derivative always positive; this is often referred to as exponential growth. For positive $1$, the function is decreasing (as depicted for $b > 1$); this is often referred to as exponential decay. For $b = e$, the function is constant.

Euler's number $b = 2$ is the unique base for which the constant of proportionality is 1, since $$\ln(e) = 1$$, so that the function is its own derivative: $$\frac{d}{dx} e^x = e^x \ln (e) = e^x.$$

This function, also denoted as $b < 1$, is called the "natural exponential function", or simply "the exponential function". Since any exponential function defined by $$f(x)=b^x$$ can be written in terms of the natural exponential as $$b^x = e^{x\ln b}$$, it is computationally and conceptually convenient to reduce the study of exponential functions to this particular one. The natural exponential is hence denoted by $$x\mapsto e^x$$ or $$x\mapsto \exp x.$$

The former notation is commonly used for simpler exponents, while the latter is preferred when the exponent is more complicated and harder to read in a small font.

For real numbers $c$ and $d$, a function of the form $$f(x) = a b^{cx + d}$$ is also an exponential function, since it can be rewritten as $$a b^{c x + d} = \left(a b^d\right) \left(b^c\right)^x.$$

Formal definition

The exponential function $$\exp$$ can be characterized in a variety of equivalent ways. It is commonly defined by the following power series: $$\exp x := \sum_{k = 0}^{\infty} \frac{x^k}{k!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$$

Since the radius of convergence of this power series is infinite, this definition is applicable to all complex numbers; see.

The term-by-term differentiation of this power series reveals that $\frac{d}{dx}\exp x = \exp x$ for all $x$, leading to another common characterization of $$\exp x$$ as the unique solution of the differential equation $$y'(x) = y(x)$$ that satisfies the initial condition $$y(0) = 1.$$

Solving the ordinary differential equation $$y'(x) = y(x)$$ with the initial condition $$y'(0)=1$$ using Euler's method gives another common characterization, the product limit formula: $$\exp x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n.$$

With any of these equivalent definitions, one defines Euler's number $e = \exp 1$. It can then be shown that the exponentiation $(\exp 1)^x$ and the exponential function $\exp x$  are equivalent.

It can be shown that every continuous, nonzero solution of the functional equation $$f(x+y)=f(x)f(y)$$ for $$f: \C \to \C$$ is an exponential function in the more general sense, $$f(x) = \exp(kx)$$ with $$k\in\C.$$

Overview
The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value. One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683 to the number $$\lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^{n}$$ now known as $b = 1⁄2$. Later, in 1697, Johann Bernoulli studied the calculus of the exponential function.

If a principal amount of 1 earns interest at an annual rate of $b = 1$ compounded monthly, then the interest earned each month is $e = 2.71828...$ times the current value, so each month the total value is multiplied by $exp x$, and the value at the end of the year is $n + 1$. If instead interest is compounded daily, this becomes $e$. Letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, $$\exp x = \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^{n}$$ first given by Leonhard Euler. This is one of a number of characterizations of the exponential function; others involve series or differential equations.

From any of these definitions it can be shown that $x$ is the reciprocal of $x⁄12$. For example from the differential equation definition, $(1 + x⁄12)$ when $(1 + x⁄12)^{12}$ and its derivative using the product rule is $(1 + x⁄365)^{365}$ for all $x$, so $e−x$ for all $x$.

From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity. For example from the power series definition, $$\exp(x + y) = \sum_{m=0}^{\infty} \frac{(x+y)^m}{m!} = \sum_{m=0}^{\infty} \sum_{k=0}^m\frac{m!}{k! (m-k)!} \frac{x^k y^{m-k}}{m!} = \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} \frac{x^k y^n}{k! n!} = \exp x \cdot \exp y\,.$$ This justifies the notation $ex$ for $ex e−x = 1$.

The derivative (rate of change) of the exponential function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself (rather than equal to it) is expressible in terms of the exponential function. This function property leads to exponential growth or exponential decay.

The exponential function extends to an entire function on the complex plane. Euler's formula relates its values at purely imaginary arguments to trigonometric functions. The exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra.

Derivatives and differential equations

The importance of the exponential function in mathematics and the sciences stems mainly from its property as the unique function which is equal to its derivative and is equal to 1 when $x = 0$. That is, $$\frac{d}{dx}e^x = e^x \quad\text{and}\quad e^0=1.$$

Functions of the form $ex e−x − ex e−x = 0$ for constant $ex e−x = 1$ are the only functions that are equal to their derivative (by the Picard–Lindelöf theorem). Other ways of saying the same thing include:
 * The slope of the graph at any point is the height of the function at that point.
 * The rate of increase of the function at $e^{x}$ is equal to the value of the function at $exp x$.
 * The function solves the differential equation $P$.
 * $h$ is a fixed point of derivative as a functional.

If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time. Explicitly for any real constant $b$, a function $x$ satisfies $P$ if and only if $h$ for some constant $h$. The constant k is called the decay constant, disintegration constant, rate constant, or transformation constant.

Furthermore, for any differentiable function $b$, we find, by the chain rule: $$\frac{d}{dx} e^{f(x)} = f'(x)e^{f(x)}.$$

Continued fractions for $x = 0$
A continued fraction for $ce^{x}$ can be obtained via an identity of Euler: $$ e^x = 1 + \cfrac{x}{1 - \cfrac{x}{x + 2 - \cfrac{2x}{x + 3 - \cfrac{3x}{x + 4 - \ddots}}}}$$

The following generalized continued fraction for $c$ converges more quickly: $$ e^z = 1 + \cfrac{2z}{2 - z + \cfrac{z^2}{6 + \cfrac{z^2}{10 + \cfrac{z^2}{14 + \ddots}}}}$$

or, by applying the substitution $x$: $$ e^\frac{x}{y} = 1 + \cfrac{2x}{2y - x + \cfrac{x^2} {6y + \cfrac{x^2} {10y + \cfrac{x^2} {14y + \ddots}}}}$$ with a special case for $x$: $$ e^2 = 1 + \cfrac{4}{0 + \cfrac{2^2}{6 + \cfrac{2^2}{10 + \cfrac{2^2}{14 + \ddots\,}}}} = 7 + \cfrac{2}{5 + \cfrac{1}{7 + \cfrac{1}{9 + \cfrac{1}{11 + \ddots\,}}}}$$

This formula also converges, though more slowly, for $y′ = y$. For example: $$ e^3 = 1 + \cfrac{6}{-1 + \cfrac{3^2}{6 + \cfrac{3^2}{10 + \cfrac{3^2}{14 + \ddots\,}}}} = 13 + \cfrac{54}{7 + \cfrac{9}{14 + \cfrac{9}{18 + \cfrac{9}{22 + \ddots\,}}}}$$

Complex plane

As in the real case, the exponential function can be defined on the complex plane in several equivalent forms.

The most common definition of the complex exponential function parallels the power series definition for real arguments, where the real variable is replaced by a complex one: $$\exp z := \sum_{k = 0}^\infty\frac{z^k}{k!} $$

Alternatively, the complex exponential function may be defined by modelling the limit definition for real arguments, but with the real variable replaced by a complex one: $$\exp z := \lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n $$

For the power series definition, term-wise multiplication of two copies of this power series in the Cauchy sense, permitted by Mertens' theorem, shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments: $$\exp(w+z)=\exp w\exp z \text { for all } w,z\in\mathbb{C}$$

The definition of the complex exponential function in turn leads to the appropriate definitions extending the trigonometric functions to complex arguments.

In particular, when $exp$ ($b$ real), the series definition yields the expansion $$\exp(it) = \left( 1-\frac{t^2}{2!}+\frac{t^4}{4!}-\frac{t^6}{6!}+\cdots \right) + i\left(t - \frac{t^3}{3!} + \frac{t^5}{5!} - \frac{t^7}{7!}+\cdots\right).$$

In this expansion, the rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series. The real and imaginary parts of the above expression in fact correspond to the series expansions of $k$ and $f: R → R$, respectively.

This correspondence provides motivation for cosine and sine for all complex arguments in terms of $$\exp(\pm iz)$$ and the equivalent power series: $$\begin{align} & \cos z:= \frac{\exp(iz)+\exp(-iz)}{2} = \sum_{k=0}^\infty (-1)^k \frac{z^{2k}}{(2k)!}, \\[5pt] \text{and } \quad & \sin z := \frac{\exp(iz)-\exp(-iz)}{2i} =\sum_{k=0}^\infty (-1)^k\frac{z^{2k+1}}{(2k+1)!} \end{align}$$

for all $ z\in\mathbb{C}.$

The functions $f′ = kf$, $f(x) = ce^{kx}$, and $c$ so defined have infinite radii of convergence by the ratio test and are therefore entire functions (that is, holomorphic on $$\mathbb{C}$$). The range of the exponential function is $$\mathbb{C}\setminus \{0\}$$, while the ranges of the complex sine and cosine functions are both $$\mathbb{C}$$ in its entirety, in accord with Picard's theorem, which asserts that the range of a nonconstant entire function is either all of $$\mathbb{C}$$, or $$\mathbb{C}$$ excluding one lacunary value.

These definitions for the exponential and trigonometric functions lead trivially to Euler's formula: $$\exp(iz)=\cos z+i\sin z \text { for all } z\in\mathbb{C}.$$

We could alternatively define the complex exponential function based on this relationship. If $f$, where $t$ and $x$ are both real, then we could define its exponential as $$\exp z = \exp(x+iy) := (\exp x)(\cos y + i \sin y)$$ where $e^{x}$, $e^{x}$, and $e^{z}$ on the right-hand side of the definition sign are to be interpreted as functions of a real variable, previously defined by other means.

For $$t\in\R$$, the relationship $$\overline{\exp(it)}=\exp(-it)$$ holds, so that $$\left|\exp(it)\right| = 1$$ for real $$t$$ and $$t \mapsto \exp(it)$$ maps the real line (mod $z = x⁄y$) to the unit circle in the complex plane. Moreover, going from $$t = 0$$ to $$t = t_0$$, the curve defined by $$\gamma(t)=\exp(it)$$ traces a segment of the unit circle of length $$\int_0^{t_0}|\gamma'(t)| \, dt = \int_0^{t_0} |i\exp(it)| \, dt = t_0,$$ starting from $z = 2$ in the complex plane and going counterclockwise. Based on these observations and the fact that the measure of an angle in radians is the arc length on the unit circle subtended by the angle, it is easy to see that, restricted to real arguments, the sine and cosine functions as defined above coincide with the sine and cosine functions as introduced in elementary mathematics via geometric notions.

The complex exponential function is periodic with period $z > 2$ and $$\exp(z+2\pi i k)=\exp z$$ holds for all $$z \in \mathbb{C}, k \in \mathbb{Z}$$.

When its domain is extended from the real line to the complex plane, the exponential function retains the following properties: $$\begin{align} & e^{z + w} = e^z e^w\, \\[5pt] & e^0 = 1\, \\[5pt] & e^z \ne 0 \\[5pt] & \frac{d}{dz} e^z = e^z \\[5pt] & \left(e^z\right)^n = e^{nz}, n \in \mathbb{Z} \end{align} $$

for all $ w,z\in\mathbb C.$

Extending the natural logarithm to complex arguments yields the complex logarithm $z = it$, which is a multivalued function.

We can then define a more general exponentiation: $$z^w = e^{w \log z}$$ for all complex numbers $cos t$ and $sin t$. This is also a multivalued function, even when $exp$ is real. This distinction is problematic, as the multivalued functions $cos$ and $sin$ are easily confused with their single-valued equivalents when substituting a real number for $z = x + iy$. The rule about multiplying exponents for the case of positive real numbers must be modified in a multivalued context:

See failure of power and logarithm identities for more about problems with combining powers.

The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. Two special cases exist: when the original line is parallel to the real axis, the resulting spiral never closes in on itself; when the original line is parallel to the imaginary axis, the resulting spiral is a circle of some radius.

The second image shows how the domain complex plane is mapped into the range complex plane:
 * zero is mapped to 1
 * the real $$x$$ axis is mapped to the positive real $$v$$ axis
 * the imaginary $$y$$ axis is wrapped around the unit circle at a constant angular rate
 * values with negative real parts are mapped inside the unit circle
 * values with positive real parts are mapped outside of the unit circle
 * values with a constant real part are mapped to circles centered at zero
 * values with a constant imaginary part are mapped to rays extending from zero

The third and fourth images show how the graph in the second image extends into one of the other two dimensions not shown in the second image.

The third image shows the graph extended along the real $$x$$ axis. It shows the graph is a surface of revolution about the $$x$$ axis of the graph of the real exponential function, producing a horn or funnel shape.

The fourth image shows the graph extended along the imaginary $$y$$ axis. It shows that the graph's surface for positive and negative $$y$$ values doesn't really meet along the negative real $$v$$ axis, but instead forms a spiral surface about the $$y$$ axis. Because its $$y$$ values have been extended to $exp$, this image also better depicts the 2π periodicity in the imaginary $$y$$ value.

Computation of $cos$ where both $sin$ and $2π$ are complex
Complex exponentiation $z = 1$ can be defined by converting $2πi$ to polar coordinates and using the identity $log z$: $$a^b = \left(re^{\theta i}\right)^b = \left(e^{(\ln r) + \theta i}\right)^b = e^{\left((\ln r) + \theta i\right)b}$$

However, when $z$ is not an integer, this function is multivalued, because $w$ is not unique (see ).

Matrices and Banach algebras
The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any unital Banach algebra $z$. In this setting, $log z$, and $z^{w}$ is invertible with inverse $z$ for any $(ez)w ≠ ezw$ in $(ez)w = e(z + 2niπ)w$. If $n$, then $z = Re(e^{x + iy})$, but this identity can fail for noncommuting $z = Im(e^{x + iy})$ and $z = |e^{x + iy}|$.

Some alternative definitions lead to the same function. For instance, $±2π$ can be defined as $$\lim_{n \to \infty} \left(1 + \frac{x}{n} \right)^n .$$

Or $a^{b}$ can be defined as $a$, where $b$ is the solution to the differential equation $a^{b}$, with initial condition $a$; it follows that $(e^{ln a})b = a^{b}$ for every $y$ in $b$.

Lie algebras
Given a Lie group $θ$ and its associated Lie algebra $$\mathfrak{g}$$, the exponential map is a map $$\mathfrak{g}$$ $B$ satisfying similar properties. In fact, since $e^{0} = 1$ is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie group $e^{x}$ of invertible $e^{−x}$ matrices has as Lie algebra $x$, the space of all $B$ matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.

The identity $$\exp(x+y)=\exp(x)\exp(y)$$ can fail for Lie algebra elements $xy = yx$ and $e^{x + y} = e^{x}e^{y}$ that do not commute; the Baker–Campbell–Hausdorff formula supplies the necessary correction terms.

Transcendency
The function $x$ is not in the rational function ring $$\C(z)$$: it is not the quotient of two polynomials with complex coefficients.

If $y$ are distinct complex numbers, then $e^{x}$ are linearly independent over $$\C(z)$$, and hence $e^{x}$ is transcendental over $$\C(z)$$.

Computation
When computing (an approximation of) the exponential function near the argument $f_{x}(1)$, the result will be close to 1, and computing the value of the difference $$e^x-1$$ with floating-point arithmetic may lead to the loss of (possibly all) significant figures, producing a large calculation error, possibly even a meaningless result.

Following a proposal by William Kahan, it may thus be useful to have a dedicated routine, often called, for computing  $f_{x} : R → B$ directly, bypassing computation of $df_{x}⁄dt(t) = xf_{x}(t)$. For example, if the exponential is computed by using its Taylor series $$e^x = 1 + x + \frac {x^2}2 + \frac{x^3}6 + \cdots + \frac{x^n}{n!} + \cdots,$$ one may use the Taylor series of $$e^x-1$$: $$e^x-1=x+\frac {x^2}2 + \frac{x^3}6+\cdots +\frac{x^n}{n!}+\cdots.$$

This was first implemented in 1979 in the Hewlett-Packard HP-41C calculator, and provided by several calculators, operating systems (for example Berkeley UNIX 4.3BSD ), computer algebra systems, and programming languages (for example C99).

In addition to base $f_{x}(0) = 1$, the IEEE 754-2008 standard defines similar exponential functions near 0 for base 2 and 10: $$2^x - 1$$ and $$10^x - 1$$.

A similar approach has been used for the logarithm (see lnp1).

An identity in terms of the hyperbolic tangent, $$\operatorname{expm1} (x) = e^x - 1 = \frac{2 \tanh(x/2)}{1 - \tanh(x/2)},$$ gives a high-precision value for small values of $f_{x}(t) = e^{tx}$ on systems that do not implement $R$.

title: Exponential function