ERFC function

Note: This draft page is under construction 🚧

Overview

ERFC (ERror Function Complementary) is a function of the Engineering category that calculates a value for the complementary error function, defined by $\text{erfc}(x)=1-\text{erf}(x)$. Also known as the complementary Gauss error function, the complementary error function represents the probability of a random variable falling outside a certain range, given that it follows a specified normal distribution.

Usage

Syntax

ERFC(X) => erfc

Argument descriptions

• X (number, required). The lower integration limit to be used to calculate the complementary error function. ERFC integrates over the range [X, $\mathrm{\infty }$).

Additional guidance

None.

Returned value

ERFC returns a number that is the complementary error function probability for the specified argument. The returned value lies in range [0, 2].

Error conditions

• In common with many other IronCalc functions, ERFC propagates errors that are found in its argument.

• If no argument, or more than one argument, is supplied, then ERFC returns the #ERROR! error.

• If the value of any argument is not (or cannot be converted to) a number, then ERFC returns the #VALUE! error.

• For some argument values, ERFC may return the #DIV/0! error.

• For more information about the different types of errors that you may encounter when using IronCalc functions, visit our Error Types page.

Details

• The complementary error function arises in many scientific, engineering, and mathematical fields and is commonly defined by the following equation (applicable for any real number $x$):

$$\text{erfc}(x)=\frac{2}{\sqrt{\pi }}{\int }_x^{\mathrm{\infty }}{e}^{-t^2}dt$$

• The figure below illustrates the output of the ERFC function for values of $x$ in the range -3 to +3.

• This figure illustrates some of the key characteristics of the complementary error function:

  • $\text{erfc}(0)=1$
  • $\text{erfc}(-x)=2-\text{erfc}(x)$
  • As $x\to \mathrm{\infty }$, $\text{erfc}(x)\to 0$
  • As $x\to -\mathrm{\infty }$, $\text{erfc}(x)\to 2$

• The complementary error function is a transcendental, non-algebraic mathematical function. IronCalc implements the ERFC function by numerical approximation using a power series.

Examples

See some examples in IronCalc.

Links

• See also IronCalc's ERF, ERF.PRECISE and ERFC.PRECISE functions.
• Visit Microsoft Excel's ERFC function page.
• Both Google Sheets and LibreOffice Calc provide versions of the ERFC function.