the above equation tells us something about complex logarithms by relating natural logarithms to imaginary (complex) numbers. Bernoulli, however, did not evaluate the integral.
Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand complex logarithms. Euler also suggested that the complex logarithms can have infinitely many values.
Cotes missed the fact that a complex logarithm can have infinitely many values, differing by multiples of 2iπ, due to the periodicity of the trigonometric functions.
Around 1740 Euler turned his attention to the exponential function instead of logarithms and obtained the formula used today that is named after him. It was published in 1748, obtained by comparing the series expansions of the exponential and trigonometric expressions.
The original proof is based on the Taylor series expansions of the exponential functionez (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 x.
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 simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number z = x + iy, and its complex conjugate, z = x − iy, can be written as
φ is the argument of z, i.e., the angle between the x axis and the vector z measured counterclockwise in radians, which is defined up to addition of 2π. Many texts write φ = tan−1y/x instead of φ = atan2(y,x), but the first equation needs adjustment when x ≤ 0. This is because for any real x and y not both zero the angles of the vectors (x, y) and (−x, −y) differ by π radians, but have the identical value of tan φ = y/x.
Use of the formula to define the logarithm of complex numbers
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):
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
Relationship between sine, cosine and exponential function
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 real-valued. For example:
Another technique is to represent the sinusoids in terms of the real part of a 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 eix is often used to simplify solutions, even if the final answer is a real function involving sine and cosine. The reason for this is that the exponential function is the eigenfunction of the operation of differentiation.
In electrical engineering, signal processing, and similar fields, signals that vary periodically over time are often described as a combination of sinusoidal functions (see Fourier analysis), and these are more conveniently expressed as the sum 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.
The exponential function ex 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 ez 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 three following definitions, which are equivalent. From a more advanced perspective, each of these definitions may be interpreted as giving the unique analytic continuation of ex to the complex plane.
Another proof is based on the fact that all complex numbers can be expressed in polar coordinates. Therefore, for somer and θ depending on x,
No assumptions are being made about r and θ; they will be determined in the course of the proof. From any of the definitions of the exponential function it can be shown that the derivative of eix is ieix. Therefore, differentiating both sides gives
Substituting r(cos θ + i sin θ) for eix and equating real and imaginary parts in this formula gives dr/dx = 0 and dθ/dx = 1. Thus, r is a constant, and θ is x + C for some constant C. The initial values r(0) = 1 and θ(0) = 0 come from e0i = 1, giving r = 1 and θ = x. This proves the formula
^Moskowitz, Martin A. (2002). A Course in Complex Analysis in One Variable. World Scientific Publishing Co. p. 7. ISBN981-02-4780-X.
^Feynman, Richard P. (1977). The Feynman Lectures on Physics, vol. I. Addison-Wesley. p. 22-10. ISBN0-201-02010-6.
^Bernoulli, Johann (1702). "Solution d'un problème concernant le calcul intégral, avec quelques abrégés par rapport à ce calcul" [Solution of a problem in integral calculus with some notes relating to this calculation]. Mémoires de l'Académie Royale des Sciences de Paris. 1702: 197–289.