Differentiation Under the Integral Sign and Error Functions
December 8, 2016
Categorised in: 1st Year Maths 2
Rule I: Integral with limits (a, b) as constants
If I(α) = $∫↙{a}↖b f(x, α) dx, then {dI}/{dα} = ∫↙{a}↖b {δ}/{δα} f(x, α) dx$
Rule II: Integral with limits as functions of parameter: Leibnitz’s Rule
${dI}/{dα} = d/{dα}∫↙{a(α)}↖{b(α)} f(x, α) dx = ∫↙{a(α)}↖{b(α)} δ/{δα} f(x, α) dx + f(b, α) {db}/{dα} -f (a,α) {da}/{dα}$
Error function
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
$erf(x) = 1/{√π} ∫↙{0}↖{x^2} e^{-t} t^{- 1/2} dt$
Properties of error functions
(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
$erfc(x) = 1/{√π} ∫↙{x^2}↖∞ e^{-t} t^{-1/2} dt$
Differentiation of error function
$d/{dx} erf (ax) = {2 a e^{-a^2 x^2}}/{√π}$
Integration by error function
$∫↙{0}↖t erf(ax) dx = t erf(at) + 1/{a√π} e^{-a^2 t^2} – 1/{a√π}$
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