Expc

Expc, short for the complex exponential function, is the extension of the real exponential function to complex arguments. In mathematical notation, expc(z) or exp(z) denotes e^z for a complex number z = x + i y. Using Euler's formula, e^z = e^x (cos y + i sin y). This gives the real and imaginary parts: Re(e^z) = e^x cos y and Im(e^z) = e^x sin y.

Properties and implications: Expc is an entire function, analytic on the complex plane, with derivative equal to itself (d/dz e^z = e^z). It satisfies e^{z1 + z2} = e^{z1} e^{z2} and has no zeros, so it never attains the value 0. Consequently, the complex logarithm of a nonzero number is multivalued. The image of expc is C \ {0}, and for any nonzero w there are infinitely many z with e^z = w (z = log|w| + i Arg(w), with Arg differing by multiples of 2π).

Computational notes: Expc can be evaluated via its Taylor series e^z = ∑_{n=0}^∞ z^n/n!, or via polar form e^{x}(cos y + i sin y). For large |z|, techniques may vary to maintain numerical stability.

Applications: The complex exponential is fundamental in complex analysis, differential equations, signal processing, and physics, mapping addition in the input to multiplication in the output and translating between trigonometric and exponential representations.

In programming contexts, expc is commonly implemented as the standard complex exponential function, often named exp or cexp in various languages. See also exponential function, Euler's formula, and complex logarithm.