Complex Powers

Calculator for exponentiation with complex numbers as base and/or exponent. This calculation has the form x^y = z, with x, y, z ∈ ℂ. Please specify the real part ℜ and the imaginary part ℑ of each base and exponent, if any. If one of these four values ​​is omitted, it is assumed to be 0. The result is output once as the real part and imaginary part separately, once as a complex number in the form a + bi, and finally as the absolute value of this complex number, the latter being a positive real number. Number values and the two real numbers e and pi are accepted as input values.

• Every number, except zero itself, raised to the power of 0 gives 1. The result of 0⁰ is a matter of definition; here, 1 is output as the result.
• Every real number raised to the power of an imaginary number, i.e., a complex number without a real part, results in a complex number whose absolute value is 1. This is that if the first and fourth fields above contain values ​​other than zero, the other two do not. In this case, the imaginary part in the exponent only changes the direction of the result on the imaginary plane, not its distance from zero. This does not apply to numbers with an imaginary part in the base!

Euler's Identity

• If the real number is Euler's number e as the base and the imaginary number is i times pi as the exponent, the result is -1. This is the so-called Euler identity, e^iπ=−1. It is often written as e^iπ+1=0, thus providing the five most important numbers in mathematics in a single formula, which is often called the most beautiful formula in the world.
The reason why this is true for the base e is somewhat complicated. In short, it goes like this: a complex number can be described by the length and direction of a vector emanating from the origin. Sine and cosine can be calculated from the direction of this vector. There are Taylor expansions of both sine and cosine, as well as of the exponential function e^x. The Taylor expansion of e^x can be composed of the Taylor expansions of sin(x) and cos(x) and requires an additional factor. This factor is the imaginary unit i. The equation is e^ix=cos(x) + i*sin(x), this is Euler's formula. If you now substitute π for x you get -1 for cos(π) and 0 for sin(π), so e^iπ=-1+0.

See also conversion and basic arithmetic for complex numbers.