Complex Error Function Calculator

Complex Error Function:

\[ \text{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^2} dt \]

Result:

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1. What is the Complex Error Function?

The complex error function, erf(z), extends the standard error function to complex numbers. It's defined by the integral of the Gaussian function in the complex plane and appears in various areas of physics and engineering.

2. How Does the Calculator Work?

The calculator computes the error function for complex numbers using the integral definition:

\[ \text{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^2} dt \]

Where:

\( z \) — Complex number (x + yi)
\( \pi \) — Pi constant (~3.14159)
\( e \) — Euler's number (~2.71828)

Explanation: The function is calculated through numerical integration or series expansion methods for complex arguments.

3. Importance of Complex Error Function

Details: The complex error function is essential in optics (Fresnel diffraction), heat conduction, probability theory, and solutions to differential equations in complex domains.

4. Using the Calculator

Tips: Enter the real and imaginary parts of your complex number. The calculator will compute erf(z) where z = x + yi.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between erf(z) and erfi(z)?
A: erfi(z) = -i erf(iz) is the imaginary error function, useful for real-valued results with imaginary arguments.

Q2: What are the special values of erf(z)?
A: erf(0) = 0, erf(∞) = 1, and erf(-∞) = -1. For pure imaginary numbers, erf(ix) is purely imaginary.

Q3: How is erf(z) computed numerically?
A: Common methods include Taylor series expansion, continued fractions, or numerical integration techniques.

Q4: What are applications of complex erf?
A: Used in wave propagation, quantum mechanics, and solutions to the heat equation in complex geometries.

Q5: Are there computational limitations?
A: Accuracy may decrease for very large |z| or when both real and imaginary parts are large.