Gaussian integral ( $\int_{- \infty}^{\infty} e^{- x^{2}} d x = π$ )

The Gaussian integral is

\int_{- \infty}^{\infty} e^{- a x^{2}} d x = \frac{π}{a}, a > 0.

(1)

Squaring the integral and changing to polar coordinates proves the formula.

Table of contents
- Dawson function Gaussian integral
  - Gaussian drift primitives Dawson function
- Error function Gaussian integral
  - Complementary error function Error function
    - Faddeeva function Complementary error function
      - Gaussian pole integral Faddeeva function

Dawson function ( $F_{D} (z) = e^{- z^{2}} \int_{0}^{z} e^{u^{2}} d u$ )

 1  0

The Dawson function obeys

F_{D}^{'} + 2 z F_{D} = 1

by differentiation. Its large-real-argument asymptotic expansion begins

F_{D} (z) = 1/ (2 z) + 1/ (4 z^{3}) + O (z^{- 5})

. This inverse-power behavior contrasts with the separate exponentially growing factors in its definition.

Gaussian drift primitives ( $E_{0} (z), E_{1} (z)$ )

 0  0

Define

E_{0} (z) = \int_{0}^{z} e^{- t^{2}} \int_{0}^{t} e^{u^{2}} d u d t

and

E_{1} (z) = \int_{0}^{z} e^{- t^{2}} \int_{0}^{t} e^{u^{2}} erf u d u d t

. They satisfy

(D^{2} + 2 zD) E_{0} = 1

and

(D^{2} + 2 zD) E_{1} = erf z

. The first is an even function, the second an odd function, and their real tails are respectively

\frac{1}{2} lo g ∣ z ∣ + C_{1} + o (1)

and

sgn z [\frac{1}{2} lo g ∣ z ∣ + C_{2} + o (1)]

. Their derivatives have inverse-linear tails, explaining the logarithmic matching constants in a singular perturbation.

Error function ( $erf$ )

 1  0

The error function is

erf (x) = \frac{2}{π} \int_{0}^{x} e^{- u^{2}} d u .

(1)

It is odd and tends to

\pm 1

x \to \pm \infty

Complementary error function ( $erfc z = 1 - erf z$ )

 0  0

The complementary error function avoids subtracting a quantity close to one when describing a small Gaussian integral tail. For real

x > 0

erfc x = (2/ π) \int_{x}^{\infty} e^{- t^{2}} d t

; analytic continuation defines it for complex arguments.

Faddeeva function ( $w (z) = e^{- z^{2}} erfc (- i z)$ )

 0  0

The Faddeeva function is an entire function built from the complementary error function. Its integral representation for

Im z > 0

w (z) = (iπ)^{- 1} \int_{- \infty}^{\infty} e^{- t^{2}} / (t - z) d t

, as normalized in NIST's integral representation. Its reflection identity is

w (- z) = 2 e^{- z^{2}} - w (z)

, directly from the odd function property of the error function.

Gaussian pole integral ( $G (a) = \int_{- \infty}^{\infty} e^{- s^{2} /4} / (s - a) d s$ )

 0  0

For a pole off the real axis, substitution

s = 2 t

into the Faddeeva function integral gives

G (a) = iπ w (a /2)

when

Im a > 0

, and

G (a) = - iπ w (- a /2)

when

Im a < 0

. The difference of boundary values is the residue theorem jump

2 πi e^{- a^{2} /4}

. In particular, for a lower-half-plane pole,

G (a) + 2 πi e^{- a^{2} /4} = iπ w (a /2)

, which packages a crossed residue and a nearby saddle into one smooth expression.

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The Gaussian integral refers to the integral of the function $ e^{-x^2} $ over the entire real line.

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