------

---  http://dlmf.nist.gov/8.4.E4   This  should produce zero for all complex z

expr=Gamma[0,z]-Integrate[(t)^(-1)*Exp[-t],{t,z,Infinity}]

expr=Gamma[0,z]-Integrate[(t)^(-1)*Exp[-t],{t,z,Infinity},GenerateConditions->False]

expr=Gamma[0,z]-Integrate[(t)^(-1)*Exp[-t],{t,z,Infinity},GenerateConditions->None]

expr=Hold[Gamma[0,z]-Integrate[(t)^(-1)*Exp[-t],{t,z,Infinity}]]

N[ReleaseHold[ReplaceAll[expr,{z->I}]]]

expr=Gamma[0,z]-Integrate[(t)^(-1)*Exp[-t],{t,z,Infinity}]

N[ReleaseHold[ReplaceAll[expr,{z->I}]]]

--- http://dlmf.nist.gov/10.43.E8

expr=(Integrate[Exp[+t]*(t)^(-ν)*BesselI[ν,t],{t,0,x},GenerateConditions->None])-(-Divide[Exp[+x]*(x)^(-ν+1),2*ν-1]*(BesselI[ν,x]-BesselI[ν-1,x])-Divide[(2)^(-ν+1),(2*ν-1)*Gamma[ν]])

N[ReplaceAll[expr,{Rule[x,1.5],Rule[ν,1.5]}]]

expr=(Integrate[Exp[+t]*(t)^(-ν)*BesselI[ν,t],{t,0,x},GenerateConditions->False])-(-Divide[Exp[+x]*(x)^(-ν+1),2*ν-1]*(BesselI[ν,x]-BesselI[ν-1,x])-Divide[(2)^(-ν+1),(2*ν-1)*Gamma[ν]])

N[ReplaceAll[expr,{Rule[x,1.5],Rule[ν,1.5]}]]

N[ReplaceAll[expr,{Rule[x,3/2],Rule[ν,3/2]}]]

N[ReplaceAll[expr,{x->1.5,ν->1.5}]]

N[ReplaceAll[expr,{x->3/2,ν->3/2}]]

expr=(Integrate[Exp[+t]*(t)^(-ν)*BesselI[ν,t],{t,0,x}])-(-Divide[Exp[+x]*(x)^(-ν+1),2*ν-1]*(BesselI[ν,x]-BesselI[ν-1,x])-Divide[(2)^(-ν+1),(2*ν-1)*Gamma[ν]])

N[ReplaceAll[expr,{Rule[x,1.5],Rule[ν,1.5]}]]

N[ReplaceAll[expr,{Rule[x,3/2],Rule[ν,3/2]}]]

x=3/2;ν=3/2;N[(NIntegrate[Exp[+t]*(t)^(-ν)*BesselI[ν,t],{t,0,x},WorkingPrecision->20])-(-Divide[Exp[+x]*(x)^(-ν+1),2*ν-1]*(BesselI[ν,x]-BesselI[ν-1,x])-Divide[(2)^(-ν+1),(2*ν-1)*Gamma[ν]]),20]

--- http://dlmf.nist.gov/10.43.E10

Clear[ν,t,x];expr=(Integrate[Exp[t]*(t)^(-ν)*BesselK[ν,t],{t,x,Infinity},GenerateConditions->None])-(Divide[Exp[x]*(x)^(-ν+1),2*ν-1]*(BesselK[ν,x]+BesselK[ν-1,x]))

N[ReplaceAll[expr,{Rule[x,1.5],Rule[ν,Power[E,Times[Complex[0,Rational[1,6]],Pi]]]}]]

expr=(Integrate[Exp[t]*(t)^(-ν)*BesselK[ν,t],{t,x,Infinity},GenerateConditions->False])-(Divide[Exp[x]*(x)^(-ν+1),2*ν-1]*(BesselK[ν,x]+BesselK[ν-1,x]))

N[ReplaceAll[expr,{Rule[x,1.5],Rule[ν,Power[E,Times[Complex[0,Rational[1,6]],Pi]]]}]]

expr=(Integrate[Exp[t]*(t)^(-ν)*BesselK[ν,t],{t,x,Infinity}])-(Divide[Exp[x]*(x)^(-ν+1),2*ν-1]*(BesselK[ν,x]+BesselK[ν-1,x]))

N[ReplaceAll[expr,{Rule[x,1.5],Rule[ν,Power[E,Times[Complex[0,Rational[1,6]],Pi]]]}]]

--- http://dlmf.nist.gov/11.5.E2

expr=2 (z/2)^ν/(Sqrt[Pi] Gamma[ν+1/2]) Integrate[Exp[-z t] (1+t^2)^(ν-1/2),{t,0,Infinity}]-StruveH[ν,z]+BesselY[ν,z]

N[ReplaceAll[expr,{z->1.5,ν->1.5}]]

expr=2 (z/2)^ν/(Sqrt[Pi] Gamma[ν+1/2]) Integrate[Exp[-z t] (1+t^2)^(ν-1/2),{t,0,Infinity},GenerateConditions->False]-StruveH[ν,z]+BesselY[ν,z]

N[ReplaceAll[expr,{z->1.5,ν->1.5}]]

expr=2 (z/2)^ν/(Sqrt[Pi] Gamma[ν+1/2]) Integrate[Exp[-z t] (1+t^2)^(ν-1/2),{t,0,Infinity},GenerateConditions->None]-StruveH[ν,z]+BesselY[ν,z]

N[ReplaceAll[expr,{z->1.5,ν->1.5}]]

z=3/2;ν=3/2;N[2 (z/2)^ν/(Sqrt[Pi] Gamma[ν+1/2]) NIntegrate[Exp[-z t] (1+t^2)^(ν-1/2),{t,0,Infinity},WorkingPrecision->20]-StruveH[ν,z]+BesselY[ν,z],20]

---  http://dlmf.nist.gov/18.17.E14

expr=(Divide[(x)^(α+μ)*LaguerreL[n,α+μ,x],Gamma[α+μ+n+1]])-(Integrate[Divide[(y)^α*LaguerreL[n,α,y],Gamma[α+n+1]]*Divide[(x-y)^(μ-1),Gamma[μ]],{y,0,x},GenerateConditions->None])

N[ReplaceAll[expr,{n->2,x->1.5,α->1.5,μ->1.5}]]

expr=(Divide[(x)^(α+μ)*LaguerreL[n,α+μ,x],Gamma[α+μ+n+1]])-(Integrate[Divide[(y)^α*LaguerreL[n,α,y],Gamma[α+n+1]]*Divide[(x-y)^(μ-1),Gamma[μ]],{y,0,x},GenerateConditions->None])

N[ReplaceAll[expr,{n->2,x->3/2,α->3/2,μ->3/2}]]

expr=(Divide[(x)^(α+μ)*LaguerreL[n,α+μ,x],Gamma[α+μ+n+1]])-(Integrate[Divide[(y)^α*LaguerreL[n,α,y],Gamma[α+n+1]]*Divide[(x-y)^(μ-1),Gamma[μ]],{y,0,x},GenerateConditions->None])

N[ReplaceAll[expr,{n->2,x->1.5,α->1.5,μ->1.5}]]

expr=(Divide[(x)^(α+μ)*LaguerreL[n,α+μ,x],Gamma[α+μ+n+1]])-(Integrate[Divide[(y)^α*LaguerreL[n,α,y],Gamma[α+n+1]]*Divide[(x-y)^(μ-1),Gamma[μ]],{y,0,x},GenerateConditions->None])

N[ReplaceAll[expr,{n->2,x->1.5,α->1.5,μ->1.5}]]

expr=(Divide[(x)^(α+μ)*LaguerreL[n,α+μ,x],Gamma[α+μ+n+1]])-(Integrate[Divide[(y)^α*LaguerreL[n,α,y],Gamma[α+n+1]]*Divide[(x-y)^(μ-1),Gamma[μ]],{y,0,x},GenerateConditions->None])

N[ReplaceAll[expr,{n->2,x->3/2,α->3/2,μ->3/2}]]

expr=(Divide[(x)^(α+μ)*LaguerreL[n,α+μ,x],Gamma[α+μ+n+1]])-(Integrate[Divide[(y)^α*LaguerreL[n,α,y],Gamma[α+n+1]]*Divide[(x-y)^(μ-1),Gamma[μ]],{y,0,x}])

N[ReplaceAll[expr,{n->2,x->1.5,α->1.5,μ->1.5}]]

expr=(Divide[(x)^(α+μ)*LaguerreL[n,α+μ,x],Gamma[α+μ+n+1]])-(Integrate[Divide[(y)^α*LaguerreL[n,α,y],Gamma[α+n+1]]*Divide[(x-y)^(μ-1),Gamma[μ]],{y,0,x}])

N[ReplaceAll[expr,{n->2,x->3/2,α->3/2,μ->3/2}]]

--- http://dlmf.nist.gov/10.22.E39  :: Correct answer is (1/8) (x^2) HypergeometricPFQ[{1,1},{2,2,2},-((x^2)/4)] for x>0

FunctionExpand[Sum[(-1)^(k-1) (x/2)^(2 k)/(2 k k!^2),{k,1,Infinity}]]

exprL=Integrate[(BesselJ[0,t])/t,{t,x,Infinity},GenerateConditions->False]+EulerGamma+Log[x/2]

exprR=Integrate[(1-BesselJ[0,t])/t,{t,0,x},GenerateConditions->False]

N[ReplaceAll[exprL-exprR,{x->1}]]

N[ReplaceAll[exprL-exprR,{x->2}]]

exprL=Integrate[(BesselJ[0,t])/t,{t,x,Infinity},GenerateConditions->None]+EulerGamma+Log[x/2]

exprR=Integrate[(1-BesselJ[0,t])/t,{t,0,x},GenerateConditions->None]

N[ReplaceAll[exprL-exprR,{x->1}]]

N[ReplaceAll[exprL-exprR,{x->2}]]

exprL=Integrate[(BesselJ[0,t])/t,{t,x,Infinity}]+EulerGamma+Log[x/2]

exprR=Integrate[(1-BesselJ[0,t])/t,{t,0,x}]

N[ReplaceAll[exprL-exprR,{x->1}]]

N[ReplaceAll[exprL-exprR,{x->2}]]