text: # erfc
  
  Complementary error function
  
  ## Syntax
  
  ``erfc(X)``
  ``erfc(K,X)``
  
  ## Description
  
  example
  
  ````erfc(X)` represents the complementary error function of `X`, that is,```erfc(X) = 1 - erf(X)```.```
  
  example
  
  ````erfc(K,X)` represents the iterated integral of the complementary error function of `X`, that is, `erfc(K, X) = int(erfc(K - 1, y), y, X, inf)`.```
  
  ## Examples
  
  ### Complementary Error Function for Floating-Point and Symbolic Numbers
  
  Depending on its arguments, `erfc` can return floating-point or exact symbolic results.
  
  Compute the complementary error function for these numbers. Because these numbers are not symbolic objects, you get the floating-point results:
  
  `A = [erfc(1/2), erfc(1.41), erfc(sqrt(2))]`
  ```A = 0.4795 0.0461 0.0455```
  
  Compute the complementary error function for the same numbers converted to symbolic objects. For most symbolic (exact) numbers, `erfc` returns unresolved symbolic calls:
  
  `symA = [erfc(sym(1/2)), erfc(sym(1.41)), erfc(sqrt(sym(2)))]`
  ```symA = [ erfc(1/2), erfc(141/100), erfc(2^(1/2))]```
  
  Use `vpa` to approximate symbolic results with the required number of digits:
  
  ```d = digits(10); vpa(symA) digits(d)```
  ```ans = [ 0.4795001222, 0.04614756064, 0.0455002639]```
  
  ### Error Function for Variables and Expressions
  
  For most symbolic variables and expressions, `erfc` returns unresolved symbolic calls.
  
  Compute the complementary error function for `x` and ```sin(x) + x*exp(x)```:
  
  ```syms x f = sin(x) + x*exp(x); erfc(x) erfc(f)```
  ```ans = erfc(x) ans = erfc(sin(x) + x*exp(x))```
  
  ### Complementary Error Function for Vectors and Matrices
  
  If the input argument is a vector or a matrix, `erfc` returns the complementary error function for each element of that vector or matrix.
  
  Compute the complementary error function for elements of matrix `M` and vector `V`:
  
  ```M = sym([0 inf; 1/3 -inf]); V = sym([1; -i*inf]); erfc(M) erfc(V)```
  ```ans = [ 1, 0] [ erfc(1/3), 2] ans = erfc(1) 1 + Inf*1i```
  
  Compute the iterated integral of the complementary error function for the elements of `V` and `M`, and the integer `-1`:
  
  ```erfc(-1, M) erfc(-1, V)```
  ```ans = [ 2/pi^(1/2), 0] [ (2*exp(-1/9))/pi^(1/2), 0] ans = (2*exp(-1))/pi^(1/2) Inf```
  
  ### Special Values of Complementary Error Function
  
  `erfc` returns special values for particular parameters.
  
  Compute the complementary error function for x = 0, x = ∞, and x = –∞. The complementary error function has special values for these parameters:
  
  `[erfc(0), erfc(Inf), erfc(-Inf)]`
  ```ans = 1 0 2```
  
  Compute the complementary error function for complex infinities. Use `sym` to convert complex infinities to symbolic objects:
  
  `[erfc(sym(i*Inf)), erfc(sym(-i*Inf))]`
  ```ans = [ 1 - Inf*1i, 1 + Inf*1i]```
  
  ### Handling Expressions That Contain Complementary Error Function
  
  Many functions, such as `diff` and `int`, can handle expressions containing `erfc`.
  
  Compute the first and second derivatives of the complementary error function:
  
  ```syms x diff(erfc(x), x) diff(erfc(x), x, 2)```
  ```ans = -(2*exp(-x^2))/pi^(1/2) ans = (4*x*exp(-x^2))/pi^(1/2)```
  
  Compute the integrals of these expressions:
  
  ```syms x int(erfc(-1, x), x)```
  ```ans = erf(x)```
  `int(erfc(x), x)`
  ```ans = x*erfc(x) - exp(-x^2)/pi^(1/2)```
  `int(erfc(2, x), x)`
  ```ans = (x^3*erfc(x))/6 - exp(-x^2)/(6*pi^(1/2)) +... (x*erfc(x))/4 - (x^2*exp(-x^2))/(6*pi^(1/2))```
  
  ### Plot Complementary Error Function
  
  Plot the complementary error function on the interval from -5 to 5.
  
  ```syms x fplot(erfc(x),[-5 5]) grid on```
  
  ## Input Arguments
  
  collapse all
  
  Input, specified as a symbolic number, variable, expression, or function, or as a vector or matrix of symbolic numbers, variables, expressions, or functions.
  
  Input representing an integer larger than `-2`, specified as a number, symbolic number, variable, expression, or function. This arguments can also be a vector or matrix of numbers, symbolic numbers, variables, expressions, or functions.
  
  collapse all
  
  ### Complementary Error Function
  
  The following integral defines the complementary error function:
  
  `$erfc\left(x\right)=\frac{2}{\sqrt{\pi }}\underset{x}{\overset{\infty }{\int }}{e}^{-{t}^{2}}dt=1-erf\left(x\right)$`
  
  Here `erf(x)` is the error function.
  
  ### Iterated Integral of Complementary Error Function
  
  The following integral is the iterated integral of the complementary error function:
  
  `$erfc\left(k,x\right)=\underset{x}{\overset{\infty }{\int }}erfc\left(k-1,y\right)dy$`
  
  Here, $erfc\left(0,x\right)=erfc\left(x\right)$.
  
  ## Tips
  
  • Calling `erfc` for a number that is not a symbolic object invokes the MATLAB® `erfc` function. This function accepts real arguments only. If you want to compute the complementary error function for a complex number, use `sym` to convert that number to a symbolic object, and then call `erfc` for that symbolic object.
  
  • For most symbolic (exact) numbers, `erfc` returns unresolved symbolic calls. You can approximate such results with floating-point numbers using `vpa`.
  
  • At least one input argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a matrix, then `erfc` expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.
  
  ## Algorithms
  
  The toolbox can simplify expressions that contain error functions and their inverses. For real values `x`, the toolbox applies these simplification rules:
  
  • ```erfinv(erf(x)) = erfinv(1 - erfc(x)) = erfcinv(1 - erf(x)) = erfcinv(erfc(x)) = x```
  
  • ```erfinv(-erf(x)) = erfinv(erfc(x) - 1) = erfcinv(1 + erf(x)) = erfcinv(2 - erfc(x)) = -x```
  
  For any value `x`, the system applies these simplification rules:
  
  • `erfcinv(x) = erfinv(1 - x)`
  
  • `erfinv(-x) = -erfinv(x)`
  
  • `erfcinv(2 - x) = -erfcinv(x)`
  
  • `erf(erfinv(x)) = erfc(erfcinv(x)) = x`
  
  • `erf(erfcinv(x)) = erfc(erfinv(x)) = 1 - x`
  
  ## References
  
  [1] Gautschi, W. “Error Function and Fresnel Integrals.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.
  
  ## Version History
  
  Introduced in R2011b