One equation to bring them all and in the darkness bind them. In the land of Euler (that would be Switzerland, Biatch!), where the Shadows (of the Alps) lie. 
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x,
History
It was Bernoulli [1702] who noted thatMeanwhile, Roger Cotes, in 1714, discovered
It was Euler (presumably around 1740) who turned his attention to the exponential function instead of logarithms, and obtained the correct formula now coined after his name. It was published in 1748, and his proof was based on the infinite series of both sides being equal. Neither of these men saw the geometrical interpretation of the formula: the view of complex numbers as points in the complex plane arose only some 50 years later (see Caspar Wessel).
Applications in complex number theory
This formula can be interpreted as saying that the function e^{ix} traces out the unit circle in the complex number plane as x ranges through the real numbers. Here, x is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in radians.
The original proof is based on the Taylor series expansions of the exponential function e^{z} (where z is a complex number) and of sin x and cos x for real numbers x (see below). In fact, the same proof shows that Euler's formula is even valid for all complex numbers z.
A point in the complex plane can be represented by a complex number written in cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates. The polar form reduces the number of terms from two to one, which simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number z = x + iy can be written as
is the argument of z—i.e., the angle between the x axis and the vector z measured counterclockwise and in radians—which is defined up to addition of 2π. Many texts write tan^{1}(y/x) instead of atan2(y,x) but this needs adjustment when x ≤ 0.
Now, taking this derived formula, we can use Euler's formula to define the logarithm of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation) that
Therefore, one can write:
Finally, the other exponential law
Relationship to trigonometry
Euler's formula provides a powerful connection between analysis and trigonometry, and provides an interpretation of the sine and cosine functions as weighted sums of the exponential function:These formulas can even serve as the definition of the trigonometric functions for complex arguments x. For example, letting x = iy, we have:
After the manipulations, the simplified result is still realvalued. For example:
Other applications
In differential equations, the function e^{ix} is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. The reason for this is that the complex exponential is the eigenfunction of differentiation. Euler's identity is an easy consequence of Euler's formula.In electrical engineering and other fields, signals that vary periodically over time are often described as a combination of sine and cosine functions (see Fourier analysis), and these are more conveniently expressed as the real part of exponential functions with imaginary exponents, using Euler's formula. Also, phasor analysis of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor.
Definitions of complex exponentiation
Main articles: Exponentiation and Exponential function
The exponential function e^{x} for real values of x may be defined in a few different equivalent ways (see Characterizations of the exponential function). Several of these methods may be directly extended to give definitions of e^{z} for complex values of z simply by substituting z in place of x and using the complex algebraic operations. In particular we may use either of the two following definitions which are equivalent. From a more advanced perspective, each of these definitions may be interpreted as giving the unique analytic continuation of e^{x} to the complex plane.
Power series definition
For complex zLimit definition
For complex zProofs
Various proofs of the formula are possible.Using power series
Here is a proof of Euler's formula using power series expansions as well as basic facts about the powers of i:Using calculus
Treating i as a constant, albeit an imaginary constant, note thatUsing differential equations
Here is another proof that follows from the differential identity above. Define a new function ƒ(x) of the real variable x asSee also
References
 ^ Moskowitz, Martin A. (2002). A Course in Complex Analysis in One Variable. World Scientific Publishing Co.. pp. 7. ISBN 981024780X.
 ^ Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. AddisonWesley. pp. 22–10. ISBN 0201020106.
 ^ Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. AddisonWesley. pp. 22–1. ISBN 0201020106.
 ^ John Stillwell (2002). Mathematics and Its History. Springer.
External links
 Proof of Euler's Formula by Julius O. Smith III
 Euler's Formula and Fermat's Last Theorem
 Complex Exponential Function Module by John H. Mathews
 Elements of Algebra
 Visual Representation of Euler's Formula
EULER'S IDENTITY

In analytical mathematics, Euler's Identity, named for the SwissGerman mathematician Leonhard Euler, is the equality
 is Euler's number, the base of natural logarithms,
 is the imaginary unit, which satisfies i^{2} = −1, and
 is pi, the ratio of the circumference of a circle to its diameter.
Beauty
Euler's identity is considered by many to be remarkable for its mathematical beauty. These three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants: The number 0, the additive identity.
 The number 1, the multiplicative identity.
 The number π, which is ubiquitous in trigonometry, the geometry of Euclidean space, and analytical mathematics (π = 3.14159265...)
 The number e, the base of natural logarithms, which occurs widely in mathematical and scientific analysis (e = 2.718281828...). Both π and e are transcendental numbers.
 The number i, the imaginary unit of the complex numbers, a field of numbers that contains the roots of all polynomials (that are not constants), and whose study leads to deeper insights into many areas of algebra and calculus, such as integration in calculus.
A poll of readers conducted by The Mathematical Intelligencer magazine named Euler's Identity as the "most beautiful theorem in mathematics".^{[1]} Another poll of readers that was conducted by Physics World magazine, in 2004, chose Euler's Identity tied with Maxwell equations (of electromagnetism) as the "greatest equation ever".^{[2]}
An entire 400page mathematics book, Dr. Euler's Fabulous Formula (published in 2006), written by Dr. Paul Nahin (a Professor Emeritus at the University of New Hampshire), is devoted to Euler's Identity. This monograph states that Euler's Identity sets "the gold standard for mathematical beauty."^{[3]}
Constance Reid claimed that Euler's Identity was "the most famous formula in all mathematics."^{[4]}
The mathematician Carl Friedrich Gauss was reported to have commented that if this formula was not immediately apparent to a student upon being told it, that student would never become a firstclass mathematician.^{[5]}
After proving Euler's Identity during a lecture, Benjamin Peirce, a noted American 19th century philosopher/mathematician and a professor at Harvard University, stated that "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth." ^{[6]}
The Stanford University mathematics professor, Dr. Keith Devlin, said, "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence."^{[7]}
Derivation
The identity is a special case of Euler's formula from complex analysis, which states that
Generalizations
Euler's Identity is actually a special case of the more general identity that the nth roots of unity, for n > 1, add up to 0:In another field of mathematics, by using quaternion exponentiation, one can show that a similar identity also applies to quaternions:
Attribution
While Euler wrote about his formula that relates e with cosine and sine terms, in the field of complex numbers, there is no known record of Euler's actually stating or deriving the simplified identity equation itself.Furthermore, Euler's formula was probably known before the life of Euler.^{[8]} (If so, then this usage would be an example of Stigler's law of eponymy.) Thus, the question of whether or not this identity should be attributed to Euler is unanswerable.
See also
Notes
References
 Crease, Robert P., "The greatest equations ever", PhysicsWeb, October 2004 (registration required).
 Crease, Robert P. "Equations as icons," PhysicsWeb, March 2007 (registration required).
 Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (New York: Penguin, 2004).
 Kasner, E., and Newman, J., Mathematics and the Imagination (Bell and Sons, 1949).
 Maor, Eli, e: The Story of a number (Princeton University Press, 1998), ISBN 0691058547
 Nahin, Paul J., Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills (Princeton University Press, 2006), ISBN 9780691118222
 Reid, Constance, From Zero to Infinity (Mathematical Association of America, various editions).
 Sandifer, Ed, "Euler's Greatest Hits", MAA Online, February 2007.
8 comments:
Some of my regular readers may be offended I make fun of Switzerland in my comments under the picture of the Titanium Euler's Identity ring caption above.
Let me assure them that first, I consider it to be gentle fun. I certainly feel no malice to the lovely Swiss.
Living in America where I live, I have met many people from the Germanic counties of central Europe, meaning Germany, Austria, and Switzerland. Each and every one of them to the last man and women are simply outstanding people, and in my humble opinion add greatly to the everchanging melting pot that is the horriblynamed "United States of America."
Welcome aboard "shotzes" (whatever that means ... we speak but one language here, heh), glad to have you here.
Having said that ....
I was very distressed to read a few years that Switzerland wasn't as "neutral" as they pride themselves as being during WWII, when it was proven they held various pieces of pricey Jewish artwork confiscated by The Third Reich. That in turn makes me wonder how "neutral" these Swiss Banks are today with their "numbered" accounts.
Julian Assange says he will blow the cover off many when he (eventually) releases his hacked "bank records" of the wealthy. I won't be surprised whatever it is, and I hope that Switzerland weathers the storm, whatever it is. But I must say ...
Leonhard Euler! My GOD, what a GREAT Swiss! What a great human, and Switzerland owns him, so thanks for that.
Also, Switzerland is the most beautiful country in Europe, based on the account of one person I know who has visited all of them.
And that's not bad, hmm?
Would it be inappropriate for one to want that ring as a wedding ring?! Can you actually buy them? I'd love to get one. I got euler's formula henna tattooed onto my wrist over the summer. If I ever did get a tattoo (which will never happen) it would definitely be this equation though.
It really is just perfection.
If it's not inappropriate to you AND Dan then it's not inappropriate at all. :)
Yes, give me a bit, I'll find where to buy it, I wish a few myself.
Well there's this, but it's not the same, bear with me ....
Aoife, I found it! Click here.
Sheesh, $215 ?? Um, OK.
You can buy that ring here: http://www.titaniumringsforever.com/blog/2011/titaniumringsformulasequations
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