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Erf

Erf[z]
gives the error function .
Erf
gives the generalized error function .
  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Erf[z] is the integral of the Gaussian distribution, given by .
  • Erf is given by .
  • Erf[z] is an entire function of z with no branch cut discontinuities.
  • For certain special arguments, Erf automatically evaluates to exact values.
  • Erf can be evaluated to arbitrary numerical precision.
  • Erf automatically threads over lists.
Evaluate numerically:
Evaluate numerically:
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Evaluate for complex arguments:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Erf threads element-wise over lists:
Simple exact values are generated automatically:
Parity transformation is automatically applied:
The two-argument form gives the difference:
Erf can be applied to a power series:
Infinite arguments give symbolic results:
CDF of normal distribution:
Cumulative probabilities for values of normal random variable to lie between and :
Solution of the heat equation for piecewise-constant initial condition:
A check that the solution fulfills the heat equation:
Plot of the solution for different times:
Compose with inverse functions:
Solve a transcendental equation:
Integrals:
Integral transforms:
Erf appears in special cases of many mathematical functions:
For large arguments, intermediate values may underflow:
The error function for large real-part arguments can be very close to 1:
Very large arguments can give unevaluated results:
A neat continued fraction:
Its limit can be expressed through Erf:
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