Absolute value function: Integration (formula 12.01.21.0008)

Abs

$Mathematica Notation$

Complex Components

Abs[z]

Integration

Contour integration

http://functions.wolfram.com/12.01.21.0008.01

Input Form

ContourIntegrate[z^(m - 1)/Abs[z - k]^(2 (n + 1)), {z, Ï Exp[I Ï†], {0, 2 Pi}}] == 2*Pi*I*(UnitStep[Ï - Abs[w]]*Sum[Binomial[n + k, k]* Binomial[n + m, n - k]*((w^m*Abs[w]^(2*k))/ (Ï^2 - Abs[w]^2)^(n + k + 1)), {k, 0, n}] + UnitStep[Abs[w] - Ï]*Sum[Binomial[n + k, k]* Binomial[n + m, n - k]*(Ï^(2*(m + k))/ (Conjugate[w]^m*(Abs[w]^2 - Ï^2)^(n + k + 1))), {k, 0, n}]) /; n âˆˆ Integers && n >= 0 && m âˆˆ Integers && m >= 0 && w != 0 && Abs[w] != Ï

Standard Form

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MathML Form

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Date Added to functions.wolfram.com (modification date)

2002-12-18