Meijer G-Function

Author

Eric W. Weisstein
November 29, 2008

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

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

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

Additional Authors

Michael Trott
contour code

Notation

MeijerG[{{a,b},{e}},{{c},{d}},z]//TraditionalForm

MeijerG[{{a,b,c},{}},{{c},{d}},z]//TraditionalForm

HoldForm[MeijerG[{{},{}},{{c},{d}},z]]//TraditionalForm

HoldForm[MeijerG[{{c},{d}},{{},{}},z]]//TraditionalForm

Indices

MeijerGIndices[p_List,q_List,z_]:={Length[First[#]]&/@{q,p},Length[Flatten[#]]&/@{p,q}}

MeijerGIndices[{{a,b,c},{}},{{c},{d}}]

Special Cases

(Subscript[G, 2,2]^(1,2))(z|({{1,1}, {1,0}}))==log(z+1)

MeijerG[{{1,1},{}},{{1},{0}},z]

(Subscript[G, 2,2]^(1,2))(z|({{1,1}, {1,1}}))==(z/(z+1))

MeijerG[{{1,1},{}},{{1},{1}},z]

(Subscript[G, 0,2]^(1,0))((z/2)|({{0,(1/2)}}))==((cos((Sqrt[2]) (Sqrt[z])))/(Sqrt[π]))

MeijerG[{{},{}},{{0},{(1/2)}},(z/2)]

(Subscript[G, 1,0]^(0,1))(z|({{1-a}}))==(e^(-1/z)) (z^(-a))

MeijerG[{{1-a},{}},{{},{}},z]

(Subscript[G, 1,1]^(1,1))(z|({{1-a}, {b}}))==(z^b) (((z+1))^(-a-b)) (Gamma[a+b])

MeijerG[{{1-a},{}},{{b},{}},z]

(Subscript[G, 0,2]^(1,0))((z/2),(1/2)|({{0,(1/2)}}))==((cos(z))/(Sqrt[π]))

MeijerG[{{},{}},{{0},{(1/2)}},(1/2)z,(1/2)]

(Subscript[G, 2,0]^(0,2))(z|({{a,b}}))==π ((z^(((a-b)/2)+b-1)) csc(π (b-a)) (BesselI[a-b, (2/(Sqrt[z]))])+(z^(((b-a)/2)+a-1)) csc(π (a-b)) (BesselI[b-a, (2/(Sqrt[z]))]))

MeijerG[{{a,b},{}},{{},{}},z]

(Subscript[G, 1,2]^(1,1))(z|({{a}, {b,c}}))==(z^b) (Gamma[-a+b+1]) (Hypergeometric1F1Regularized[-a+b+1, b-c+1, -z])

MeijerG[{{a},{}},{{b},{c}},z]

(Subscript[G, 3,2]^(1,1))(z|({{(Subscript[a, 1]),(Subscript[a, 2]),(Subscript[a, 3])}, {(Subscript[b, 1]),(Subscript[b, 2])}}))==(((z^((Subscript[a, 1])-1)) (Gamma[-(Subscript[a, 1])+(Subscript[b, 1])+1]) (Subscript[, 2])(Subscript[(OverTilde[F]), 2])(-(Subscript[a, 1])+(Subscript[b, 1])+1,-(Subscript[a, 1])+(Subscript[b, 2])+1;-(Subscript[a, 1])+(Subscript[a, 2])+1,-(Subscript[a, 1])+(Subscript[a, 3])+1;(1/z)))/(Gamma[(Subscript[a, 1])-(Subscript[b, 2])]))

MeijerG[{{(Subscript[a, 1])}, {(Subscript[a, 2]), (Subscript[a, 3])}}, {{(Subscript[b, 1])}, {(Subscript[b, 2])}}, z]

(Subscript[G, 2,4]^(2,1))(z|({{(Subscript[a, 1]),(Subscript[a, 2])}, {(Subscript[b, 1]),(Subscript[b, 2]),(Subscript[b, 3]),(Subscript[b, 4])}}))==(π (z^(Subscript[b, 1])) csc(π ((Subscript[b, 2])-(Subscript[b, 1]))) (Gamma[-(Subscript[a, 1])+(Subscript[b, 1])+1]) (Subscript[, 2])(Subscript[F, 3])(-(Subscript[a, 1])+(Subscript[b, 1])+1,-(Subscript[a, 2])+(Subscript[b, 1])+1;(Subscript[b, 1])-(Subscript[b, 2])+1,(Subscript[b, 1])-(Subscript[b, 3])+1,(Subscript[b, 1])-(Subscript[b, 4])+1;-z))/((Gamma[(Subscript[b, 1])-(Subscript[b, 2])+1]) (Gamma[(Subscript[b, 1])-(Subscript[b, 3])+1]) (Gamma[(Subscript[b, 1])-(Subscript[b, 4])+1]) (Gamma[(Subscript[a, 2])-(Subscript[b, 1])]))+(π (z^(Subscript[b, 2])) csc(π ((Subscript[b, 1])-(Subscript[b, 2]))) (Gamma[-(Subscript[a, 1])+(Subscript[b, 2])+1]) (Subscript[, 2])(Subscript[F, 3])(-(Subscript[a, 1])+(Subscript[b, 2])+1,-(Subscript[a, 2])+(Subscript[b, 2])+1;-(Subscript[b, 1])+(Subscript[b, 2])+1,(Subscript[b, 2])-(Subscript[b, 3])+1,(Subscript[b, 2])-(Subscript[b, 4])+1;-z))/((Gamma[-(Subscript[b, 1])+(Subscript[b, 2])+1]) (Gamma[(Subscript[b, 2])-(Subscript[b, 3])+1]) (Gamma[(Subscript[b, 2])-(Subscript[b, 4])+1]) (Gamma[(Subscript[a, 2])-(Subscript[b, 2])]))

FunctionExpand[MeijerG[{{(Subscript[a, 1])},{(Subscript[a, 2])}},{{(Subscript[b, 1]),(Subscript[b, 2])},{(Subscript[b, 3]),(Subscript[b, 4])}},z]]

(∫_0^∞)(x^(α-1)) exp(-a (x^3)) sin(b (x^2))dx

Integrate[x^(α-1)Exp[-a x^3]Sin[b x^2],{x,0,∞}]

(∫_0^∞)(x^(α-1)) exp(-a (x^5)) sin(b (x^3))dx

Integrate[x^(α-1)Exp[-a x^5]Sin[b x^3],{x,0,∞}]

Integration Contour

This code by Michael Trott

(Subscript[G, 1,2]^(2,1))(2 z\[VerticalSeparator]({{(1/2)}, {3,-3}}))

a1=1/2;
b1=3;
b2=-3;

dd[s_]:=Block[{s1=Gamma[1-a1-s],s2=Gamma[b1+s] ,s3=Gamma[b2+s],t1,t2,t3},
		{t1,t2,t3}=Abs[Log[Abs[{s1,s2,s3}]]];
	{Log[10,Abs[s1 s2 s3]],RGBColor@@(Abs[{t1,t2,t3}]/(Abs[t1]+Abs[t2]+Abs[t3]))}]

eps=10^-9;
pic1=Show[Graphics3D[{EdgeForm[],
	Table[Partition[Graphics3D[Plot3D[dd[sr+I si],{sr,i+eps,i+1/2-eps},{si,0,3/2}, PlotPoints->{14,40},
	DisplayFunction->Identity,
		Compiled->False]][[1]],2],
	{i,-9,7,1/2}]/.rgbc_RGBColor->SurfaceColor[rgbc,rgbc,2.7]}],
BoxRatios->{5,1.4,1}];

pic2=pic1/.Polygon[l_]:>Polygon[{1,-1,1}#&/@l];

the 2d contour:

pp=6;
Show[Graphics[{Thickness[0.01],
		Line[{{3+1/4,-3/2},{3+1/4,-5/4}}],
		Line[Table[{3,-5/4}+1/4 {Cos[φ],Sin[φ]} ,{φ,0,π/2,π/2/pp}]],
		Line[{{3,-1},{1/2,-1}}],
		Line[Table[{1/2,-3/4}+1/4 {Cos[φ],Sin[φ]} ,{φ,-π/2,-π,-π/2/pp}]],
		Line[{{1/4,-3/4},{1/4,0}}],
		Line[Table[{1/2,0}+1/4 {Cos[φ],Sin[φ]} ,{φ,π,0,-π/2/pp}]],
     Line[Table[{2/2,0}+1/4 {Cos[φ],Sin[φ]} ,{φ,π,2π,π/2/pp}]],
		Line[Table[{3/2,0}+1/4 {Cos[φ],Sin[φ]} ,{φ,π,0,-π/2/pp}]],
     Line[Table[{4/2,0}+1/4 {Cos[φ],Sin[φ]} ,{φ,π,2π,π/2/pp}]],	
		Line[Table[{5/2,0}+1/4 {Cos[φ],Sin[φ]} ,{φ,π,0,-π/2/pp}]],
     Line[Table[{6/2,0}+1/4 {Cos[φ],Sin[φ]} ,{φ,π,2π,π/2/pp}]],
     Line[{{3+1/4,0},{3+1/4,5/4}}],
			
			Table[{Hue[0],PointSize[0.01],Point[{i,0}]},{i,-4,3}],
			Table[{Hue[0.22],PointSize[0.01],Point[{i,0}]},{i,1/2,9/2}]
			
			
		}],PlotRange->All, Frame->True,AspectRatio->Automatic];

tube version of the 2d contour:

torus[mp_, r1_, r2_,{f11_,f12_},{pp1_,pp2_}] :=
	Cases[ParametricPlot3D[Evaluate[mp+ {r1 Cos[f1] + r2 Cos[f1] Cos[f2],
                                                                                      r1 Sin[f1] + r2 Sin[f1] Cos[f2],
                                                                                      r2         Sin[f2]}],
		{f1,f11,f12},{f2,0,2Pi},PlotPoints->{pp1,pp2},
		DisplayFunction->Identity],
		_Polygon,Infinity]

tubez[p1_,p2_,r2_,{pp1_,pp2_}]:=	Cases[ParametricPlot3D[Evaluate[p1+t (p2-p1)+{r2 Cos[f2],0,r2 Sin[f2]}],
		{t,0,1},{f2,0,2Pi},PlotPoints->{Ceiling[12 Sqrt[(p2-p1).(p2-p1)]]+1,pp2},
		DisplayFunction->Identity],
		_Polygon,Infinity]

tubex[p1_,p2_,r2_,{pp1_,pp2_}]:=	Cases[ParametricPlot3D[Evaluate[p1+t (p2-p1)+{0,r2 Cos[f2],r2 Sin[f2]}],
		{t,0,1},{f2,0,2Pi},PlotPoints->{Ceiling[12Sqrt[12(p2-p1).(p2-p1)]]+1,pp2},
		DisplayFunction->Identity],
		_Polygon,Infinity]

pp2=17;pp1=18;r2=0.025;
theTube=Cases[
		Show[Graphics3D[{EdgeForm[Thickness[0.001]],
		tubez[{3+1/4,-3/2,0},{3+1/4,-5/4,0},r2,{2,pp2}],
		torus[{3,-5/4,0},1/4 ,r2,{0,π/2},{pp1,pp2}],
		tubex[{3,-1,0},{1/2,-1,0},r2,{2,pp2}],	
		torus[{1/2,-3/4,0},1/4 ,r2,{-π/2,-π},{pp1,pp2}],
		tubez[{1/4,-3/4,0},{1/4,0,0},r2,{2,pp2}],
		torus[{1/2,0,0},1/4 ,r2 ,{π,0},{pp1,pp2}],
		torus[{2/2,0,0},1/4 ,r2,{π,2π},{pp1,pp2}],
		torus[{3/2,0,0},1/4 ,r2 ,{π,0},{pp1,pp2}],
     torus[{4/2,0,0},1/4 ,r2 ,{π,2π},{pp1,pp2}],	
		torus[{5/2,0,0},1/4 ,r2 ,{π,0},{pp1,pp2}],
     torus[{6/2,0,0},1/4 ,r2 ,{π,2π},{pp1,pp2}],
		tubez[{3+1/4,0,0},{3+1/4,5/4,0},r2,{2,pp2}]
			
			}],PlotRange->All, Axes->True],_Polygon,∞];

ϵ=10^-2;
toSurface[{x_,y_,z_}]:={x,y,First[dd[x+I y]]+z+r2+ϵ}

pic3=Show[Graphics3D[{EdgeForm[],SurfaceColor[Hue[0.02],Hue[0.15],2.3],
				theTube/. Polygon[l_]:> Polygon[toSurface /@l]}]]

Show[{pic1,pic2,pic3},BoxRatios->{2.5,1,1}, Axes->True,ColorOutput->CMYKColor];

Not defined outside |z|=1

MeijerG[{{},{(1/2),(1/2)}},{{0,2},{}},(x^2)]/.x->1.1