SPECIAL FUNCTIONS / GAUSSIAN INTEGRAL
Evaluate erf(x), its complement, and the odd-symmetry check for a real input.
METHOD / WORKED EXAMPLE
The error function accumulates a normalized exp(-t squared) integral from zero to x. Reporting erfc and odd symmetry helps detect sign or complement mistakes in downstream formulas.
WORKED DEFAULT
For x = 1, erf(1) is approximately 0.84270079 and erfc(1) is approximately 0.15729921. Their sum is 1, while erf(-1) equals -erf(1).
READ THE RESULT
erf is bounded between -1 and 1 for real inputs. Its numerical value becomes a normal-probability expression only after using the distribution's required square-root-of-two transformation.
ASSUMPTIONS AND LIMITS
The output is a special-function approximation, not automatically a normal-distribution probability until the correct scaling and event definition are applied.
COMMON QUESTIONS
Not without scaling and interpretation. The standard normal CDF satisfies Phi(z) = one half times [1 + erf(z/sqrt(2))]. Using erf(z) directly changes the argument and gives the wrong probability. Start from the exact probability event, apply the square-root-of-two transformation, and distinguish cumulative, central, and tail areas.
The defining integral runs from zero to x. Reversing that direction for a negative x changes the sign while the Gaussian integrand remains positive, so erf is an odd function: erf(-x) = -erf(x). The displayed symmetry residual checks that property and can reveal accidental absolute-value or sign handling in another formula.
The page uses a common finite polynomial approximation in double precision, with error suitable for ordinary educational and planning calculations but not arbitrary-precision scientific work. The identity checks test internal consistency, not absolute error against an exact integral. For stringent tolerances, validate with a certified numerical library and record its version and precision settings.
RELATED TOOLS
Use boundary
Evaluate a bounded polynomial approximation to the normalized Gaussian integral, then calculate erfc and the odd-symmetry residual as consistency checks.
erf(x) = (2/sqrt(pi)) integral from 0 to x of exp(-t^2) dt.