Working definition
The error function is defined at DLMF 7.2.1 as two over the square root of pi, times the integral of exp of minus t squared from zero to z. Its complement is defined at 7.2.2 as the same normalised integral taken from z to infinity, and that equation also states the identity that erfc z equals one minus erf z. Two relatives share the chapter. The Dawson integral, at 7.2.5, is exp of minus z squared times the integral of exp of t squared from zero to z. The Fresnel integrals at 7.2.7 and 7.2.8 integrate cosine and sine of pi t squared over two. All are recorded as entire functions of their complex argument.
Notation
erf zerfc zF(z) for the Dawson integralC(z) and S(z) for the Fresnel integralsAssumptions
- The definitions are stated for complex argument, and the chapter records all of these as entire functions.
- Because the integrands are entire, the integral from zero to z does not depend on the path taken between them.
- The factor of two over the square root of pi in 7.2.1 is part of the definition; an unnormalised Gaussian integral is a different function.
- The identity in 7.2.2 relating erfc to erf holds as stated and is not an approximation.
Invariants
- erf z = (2/sqrt(pi)) times the integral of exp(-t^2) from 0 to z, DLMF 7.2.1
- erfc z = 1 - erf z, DLMF 7.2.2
- F(z) = exp(-z^2) times the integral of exp(t^2) from 0 to z, DLMF 7.2.5
Reproducible procedure
- Take the definition from 7.2.1 or 7.2.2 rather than from a normal-distribution table, so the normalisation is explicit.
- Convert between erf and erfc with the identity in 7.2.2 rather than by subtracting rounded values.
- For a tail probability, evaluate erfc directly rather than forming one minus erf, so the small quantity is never the difference of two near-equal numbers.
- Before reading a Fresnel value from another reference, check whether it uses the pi t squared over two convention of 7.2.7 and 7.2.8.
Error and boundary controls
- Computing erfc as one minus erf loses significance when erf is close to one. The complementary form at 7.2.2 exists so that the small quantity can be computed directly.
- The Dawson integral at 7.2.5 is written as a product of a decaying exponential and a growing one. The product stays bounded while the factors do not, so evaluating them separately overflows for moderate argument even though the function is well behaved.
- The Fresnel integrals at 7.2.7 and 7.2.8 use the argument convention pi t squared over two. Other references omit that factor, and reading a value under the wrong convention rescales the argument.
- These are definitions rather than algorithms, so they carry no accuracy statement for any particular evaluation method.
What this does not establish
These are definitions of integrals. They do not establish that an observed distribution is Gaussian, and no statistical interpretation follows from the definition alone.
Explicit applications
0 cross-domain bridges
This foundational concept currently supports related concepts; a direct domain bridge is scheduled for a later registry version.
Authoritative references
- [1]DLMF Chapter 7: Error Function, Dawson Integral, and Fresnel Integrals · National Institute of Standards and Technology
Establishes: Integral definitions for the error function, the complementary error function, the Dawson integral and the Fresnel integrals, and the identity relating erf to erfc.
Boundary: The chapter defines the functions and their relations. It does not establish that any numerical implementation attains a stated accuracy, and it carries no claim about statistical interpretation in an applied setting.
Direct answer
- The error function is defined at DLMF 7.2.1 as two over the square root of pi, times the integral of exp of minus t squared from zero to z. Its complement is defined at 7.2.2 as the same normalised integral taken from z to infinity, and that equation also states the identity that erfc z equals one minus erf z. Two relatives share the chapter. The Dawson integral, at 7.2.5, is exp of minus z squared times the integral of exp of t squared from zero to z. The Fresnel integrals at 7.2.7 and 7.2.8 integrate cosine and sine of pi t squared over two. All are recorded as entire functions of their complex argument.
Mechanism and method
- Take the definition from 7.2.1 or 7.2.2 rather than from a normal-distribution table, so the normalisation is explicit.
- Convert between erf and erfc with the identity in 7.2.2 rather than by subtracting rounded values.
- For a tail probability, evaluate erfc directly rather than forming one minus erf, so the small quantity is never the difference of two near-equal numbers.
- Before reading a Fresnel value from another reference, check whether it uses the pi t squared over two convention of 7.2.7 and 7.2.8.
What is measured
- erf z = (2/sqrt(pi)) times the integral of exp(-t^2) from 0 to z, DLMF 7.2.1
- erfc z = 1 - erf z, DLMF 7.2.2
- F(z) = exp(-z^2) times the integral of exp(t^2) from 0 to z, DLMF 7.2.5
Limitations
- Computing erfc as one minus erf loses significance when erf is close to one. The complementary form at 7.2.2 exists so that the small quantity can be computed directly.
- The Dawson integral at 7.2.5 is written as a product of a decaying exponential and a growing one. The product stays bounded while the factors do not, so evaluating them separately overflows for moderate argument even though the function is well behaved.
- The Fresnel integrals at 7.2.7 and 7.2.8 use the argument convention pi t squared over two. Other references omit that factor, and reading a value under the wrong convention rescales the argument.
- These are definitions rather than algorithms, so they carry no accuracy statement for any particular evaluation method.
- The definitions are stated for complex argument, and the chapter records all of these as entire functions.
- Because the integrands are entire, the integral from zero to z does not depend on the path taken between them.
- The factor of two over the square root of pi in 7.2.1 is part of the definition; an unnormalised Gaussian integral is a different function.
- The identity in 7.2.2 relating erfc to erf holds as stated and is not an approximation.
What this does not establish
- These are definitions of integrals. They do not establish that an observed distribution is Gaussian, and no statistical interpretation follows from the definition alone.