Complex exponential

The function defined for complex z by exp(z)=∑ₙ₌₀∞ zⁿ/n!. On the imaginary axis it traces the unit circle and satisfies exp(iθ)=cos θ+i sin θ.

Exponential function: The function characterized by eˣ, whose derivative equals itself and whose values convert sums into products. The complex exponential extends this familiar function from real inputs to the complex plane.
Complex number: A number of the form a+bi, where a and b are real and i²=−1. Every input and output of the complex exponential is a complex number.
Fourier transform: An integral transform that expresses a signal in terms of its frequency components. Its complex exponential kernel encodes each frequency as a rotating phase.
Real exponential function: The real-valued function eˣ defined for real x, increasing from zero without bound. Unlike the complex function on imaginary inputs, it does not trace a circle.
Taylor series: A power series that represents a function through its derivatives at a point. The defining series converges for every complex input, making the function entire.
Imaginary unit: The number i whose square is −1. Powers of i cycle through four values, underlying the series' circular behavior.
Fourier series: A representation of a periodic function as a sum of sinusoidal harmonics. Complex exponentials compactly represent the sine and cosine harmonics.
Complex logarithm: The multivalued inverse relation to the complex exponential, with values differing by integer multiples of 2πi. Its multiple branches arise because the exponential repeats every 2πi.
Euler's formula: The identity eⁱˣ = cos x + i sin x for real x. It states precisely how imaginary inputs produce circular motion.
Complex conjugate: The complex number a−bi paired with a+bi, formed by reversing the sign of the imaginary part. Conjugation commutes with the exponential: exp(conjugate z)=conjugate exp(z).

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