erfi

erfi, the imaginary error function, is an entire function defined by erfi(z) = -i erf(i z) or equivalently erfi(z) = (2/√π) ∫_0^z e^{t^2} dt. It is the analytic continuation of the real error function erf(z) obtained by replacing z with i z. For real arguments x, erfi(x) is real-valued and odd, growing rapidly as |x| increases.

A standard power series for erfi is erfi(x) = (2/√π) ∑_{n=0}^∞ x^{2n+1}/(n!(2n+1)), which converges for all complex x. The derivative satisfies erfi'(x) = (2/√π) e^{x^2}, reflecting its role as the integral of e^{t^2}.

Key properties include that erfi is an entire function with no singularities, and its growth along large |z| is governed by erfi(z) ~ e^{z^2}/(√π z) for suitable directions in the complex plane. The relation to erf is erfi(z) = -i erf(i z), linking the two through analytic continuation.

Common contexts and uses involve evaluating integrals of the form ∫ e^{t^2} dt, solving differential equations with exponential of a squared term, and applications in physics and engineering where Gaussian weights appear with opposite sign in the exponent. In such settings, erfi provides an explicit special function counterpart to erf for imaginary arguments and related calculations, distinct from but connected to Dawson-type functions and other error-function variants.