Let x be a complex variable of C\ {∞}. The function Imaginary Error Function (noted erfi) is defined by the following second-order differential equation
The reason why this function has its own name is that it is not elementary anymore; it cannot be expressed by the usual elementary functions. (By the way, this is not atypical for integrals of elementary functions.

Imaginary error function.

Compute the erfi imaginary error function.

The imaginary error function is defined as

When the results of a series of measurements are described by a normal distribution with standard deviation σ and expected value 0, then erf (a/σ √2) is the probability that the error of a single measurement lies between −a and +a, for positive a. This is useful, for example, in determining the bit error rate of a digital communication system.

The error and complementary error functions occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function.

The error function and its approximations can be used to estimate results that hold with high probability or with low probability. Given a random variable X ~ Norm[μ,σ] (a normal distribution with mean μ and standard deviation σ) and a constant L < μ:
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