Differentiation Under the Integral Sign and Error Functions

Rule I: Integral with limits (a, b) as constants

If I(α) = $∫↙{a}↖b f(x, α) dx, then {dI}/{dα} = ∫↙{a}↖b {δ}/{δα} f(x, α) dx$

${dI}/{dα} = d/{dα}∫↙{a(α)}↖{b(α)} f(x, α) dx = ∫↙{a(α)}↖{b(α)} δ/{δα} f(x, α) dx + f(b, α) {db}/{dα} -f (a,α) {da}/{dα}$

1. $erf(x) = 2/{√π} ∫↙{0}↖x e^{-u^2} du$

2. Complementary error function

$erfc(x) = 2/{√π} ∫↙{x}↖∞ e^{-u^2} du $

3. Alternate definition of error function

$erf(x) = 1/{√π} ∫↙{0}↖{x^2} e^{-t} t^{- 1/2} dt$

(1) erf(∞) = 1

(2) erf(0) = 0

(3) erf(X) + erfc(x) = 1

(4) Error function is an odd function

erf(-x) = -erf(x)

(5) Expression for erf(x) in series

erf(x) = $2/{√π} [x – x^3/3 + x^5/{10} – x^7/{42} +……]$

(6) Alternate definition of complementary error function

$erfc(x) = 1/{√π} ∫↙{x^2}↖∞ e^{-t} t^{-1/2} dt$

$d/{dx} erf (ax) = {2 a e^{-a^2 x^2}}/{√π}$

$∫↙{0}↖t erf(ax) dx = t erf(at) + 1/{a√π} e^{-a^2 t^2} – 1/{a√π}$