Marcum Q-Function

Author

Eric W. Weisstein
January 9, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/MarcumQ-Function.nb.

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

©2010 Wolfram Research, Inc. except for portions noted otherwise

Integral

MarcumQ[M_,{a_,b_},opts___]:=1/a^(M-1)Integrate[x ^M  Exp[-(x^2+a^2)/2]BesselI[M-1,a x],{x,b,∞},opts]

MarcumQ[M,{α,β}]

Usage

?MarcumQ

Special Cases

(Subscript[Q, m])(0,b)

Built-in

MarcumQ[m,0, b]

By Hand

MarcumQ[m,{0, b},Assumptions->m∈Integers]

Table[With[{m=Random[Integer,{1,1}],b=Random[Real,{0,3}]},
{(2^((1/2) (-1+m))) BesselI[-1+m,0] Gamma[((1+m)/2),((b^2)/2)],
NIntegrate[x ^m  Exp[-(x^2)/2]BesselI[m-1,0],{x,b,∞}],
GammaRegularized[m, b^2/2]}
],{5}]//ColumnForm

Table[With[{m=Random[Real,{0,1}],b=Random[Real,{0,3}]},
{(2^((1/2) (-1+m))) BesselI[-1+m,0] Gamma[((1+m)/2),((b^2)/2)],
NIntegrate[x ^m  Exp[-(x^2)/2]BesselI[m-1,0],{x,b,∞}],
GammaRegularized[m, b^2/2]}
],{5}]//ColumnForm

(Subscript[Q, 1])(a,b)

Built in

FunctionExpand[MarcumQ[1,a,b]]

By Hand

MarcumQ[1,{a,b}]//Timing

MarcumQ[1,{a,b},Assumptions->(a|b)∈Reals]//Timing

(Subscript[Q, 1])(a,a)

Built in

MarcumQ[1,a,a]

By Hand

MarcumQ[1,{a,a}]//Timing

MarcumQ[1,{a,a},Assumptions->(a|b)∈Reals]//Timing

Table[With[{a=Random[Real,{0,2}]},
{NIntegrate[(E^((1/2) (-(a^2)-(x^2)))) x BesselI[0,a x],{x,a,∞}],(1+Exp[-a^2]BesselI[0,a^2])/2}
],{5}]

Series

V8

Exp[-(a^2+b^2)/2]Sum[(a/b)^k BesselI[k,a b],{k,1-M,∞}]

Identity

FunctionExpand[MarcumQ[1,a,b]+MarcumQ[1,b,a]]

Derivatives

Built in

D[MarcumQ[M,a,b],b]

By hand

FullSimplify[D[MarcumQ[M,{a,b}],b]]

D[MarcumQ[M,{a,b}],M]

D[MarcumQ[M,{a,b}],a]

MarcumQ[1,{.01,b}]

Intentities

MarcumQ[M,{a,b}]+MarcumQ[M,{b,a}]

Table[
With[{a=Random[Real,{0,2}],b=Random[Real,{0,2}]},
{NIntegrate[(E^((1/2) (-(a^2)-(x^2)))) x BesselI[0,a x],{x,b,∞}]+NIntegrate[(E^((1/2) (-(b^2)-(x^2)))) x BesselI[0,b x],{x,a,∞}],
1+Exp[-(a^2+b^2)/2]BesselI[0,a b]}
],{10}]