
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that demonstrates the deep relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x,
where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes called cis(x). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.^{[1]}
Richard Feynman called Euler's formula "our jewel"^{[2]} and "one of the most remarkable, almost astounding, formulas in all of mathematics."^{[3]}
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It was Bernoulli [1702] who noted that
And since
the above equation tells us something about complex logarithms. Bernoulli, however, did not evaluate the integral. His correspondence with Euler (who also knew the above equation) shows that he didn't fully understand logarithms. Euler also suggested that the complex logarithms can have infinitely many values.
Meanwhile, Roger Cotes, in 1714, discovered
(where "ln" means natural logarithm, i.e. log with base e).^{[4]} We now know that the above equation is only true modulo integer multiples of 2πi, but Cotes missed the fact that a complex logarithm can have infinitely many values which owes to the periodicity of the trigonometric functions.
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).
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
where
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
and that
both valid for any complex numbers a and b.
Therefore, one can write:
for any z ≠ 0. Taking the logarithm of both sides shows that:
and in fact this can be used as the definition for the complex logarithm. The logarithm of a complex number is thus a multivalued function, because φ is multivalued.
Finally, the other exponential law
which can be seen to hold for all integers k, together with Euler's formula, implies several trigonometric identities as well as de Moivre's formula.
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:
The two equations above can be derived by adding or subtracting Euler's formulas:
and solving for either cosine or sine.
These formulas can even serve as the definition of the trigonometric functions for complex arguments x. For example, letting x = iy, we have:
Complex exponentials can simplify trigonometry, because they are easier to manipulate than their sinusoidal components. One technique is simply to convert sinusoids into equivalent expressions in terms of exponentials. After the manipulations, the simplified result is still realvalued. For example:
Another technique is to represent the sinusoids in terms of the real part of a more complex expression, and perform the manipulations on the complex expression. For example:
This formula is used for recursive generation of cos(nx) for integer values of n and arbitrary x (in radians).
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.
In general, raising e to a positive integer exponent has a simple interpretation in terms of repeated multiplication of e. Raising e to zero or a negative integer exponent can be understood as repeated division. A rational number exponent can be defined by radicals of e, and an irrational number exponent can be defined by finding rationalnumber exponents that are arbitrarily close to the irrationalnumber exponent, in a limit process. However, to define and understand a complex number exponent of e, a different type of generalization is required for the concept of exponentiation.
In fact, several definitions are possible. All of them can be proven to be welldefined and equivalent, although the proofs are not included in this article.
For any real x, the following series is equal to e^{x}:
(in other words, this is the Taylor series for the real exponential function, and it has an infinite radius of convergence). This invites the following definition of e^{z} for complex z:
This can be proven to be welldefined; in particular, the series converges for any z.
A simpletostate, equivalent definition is that e^{z}, for complex z, is the analytic continuation of the function e^{x} for real x. This can be proven to be welldefined; in particular, it yields a singlevalued function on the complex plane.
For any real x, the following limit is equal to e^{x}:
This motivates the following definition of e^{z} for complex z:
For real x, the function f(x) = e^{x} is wellknown to be the unique real function satisfying the differential equation:
for all x. This motivates a definition of f(z) = e^{z} for complex z as the function that satisfies the differential equation:
for all complex z, where the derivative in f '(z) is defined in the sense of a complex derivative. This can be proven to yield a unique function which is welldefined everywhere on the complex plane.
Various proofs of this formula are possible. The first proof below starts with the "Taylor series definition" of e^{z}, while the other two use the "differential equation definition" of e^{z} (see above).
Here is a proof of Euler's formula using Taylor series expansions as well as basic facts about the powers of i:
and so on. The functions e^{x}, cos x and sin x of the (real) variable x can be expressed using their Taylor expansions around zero:
For complex z we define each of these functions by the above series, replacing the real variable x with the complex variable z. This is possible because the radius of convergence of each series is infinite. We then find that
The rearrangement of terms is justified because each series is absolutely convergent. Taking z = x to be a real number gives the original identity as Euler discovered it.
Define the (possibly complex) function ƒ(x), of real variable x, as
The derivative of ƒ(x), according to the product rule, is:
Therefore, ƒ(x) must be a constant function in x. Because ƒ(0) is known, the constant that ƒ(x) equals for all real x is also known. Thus,
Multiplying both sides by e^{ix}, we get
Define the function g(x) by
Considering that i is constant, the first and second derivatives of g(x) are
because i ^{2} = −1 by definition. From this the following 2^{nd}order linear ordinary differential equation is constructed:
or
Being a 2^{nd}order differential equation, there are two linearly independent solutions that satisfy it:
Both cos and sin are real functions in which the 2^{nd} derivative is identical to the negative of that function. Any linear combination of solutions to a homogeneous differential equation is also a solution. Then, in general, the solution to the differential equation is
for any constants A and B. But not all values of these two constants satisfy the known initial conditions for g(x):
However these same initial conditions (applied to the general solution) are
resulting in
and, finally,
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Named after the 18th century Swiss mathematician Leonhard Euler.
Singular 
Plural 
Euler's formula


