Dirichlet's Formula

Author

Eric W. Weisstein
May 8, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/DirichletsFormula.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DirichletsFormula.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Formula

Integrate[(t-τ)^(μ-1)*(τ-x)^(ν-1)*g[τ,x],{τ,0,t},{x,0,τ}]==Integrate[Integrate[(t-τ)^(μ-1)*(τ-x)^(ν-1)*g[τ,x],{τ,x,t}],{x,0,t}]/;Re[μ]>0&&Re[ν]>0

(∫_0^t)(∫_0^τ)g(τ,x) (((t-τ))^(μ-1)) (((τ-x))^(ν-1))\[DifferentialD]x\[DifferentialD]τ==(∫_0^t)((∫_x^t)g(τ,x) (((t-τ))^(μ-1)) (((τ-x))^(ν-1))\[DifferentialD]τ)\[DifferentialD]x/;Re(μ)>0∧Re(ν)>0