# Integral invoving product of Whittaker functions

I'm trying to evaluate the integral $$\int W_{0,a}(x)\,W_{1,b}(x)\,\frac{dx}{x},$$ where $W$ denotes the Whittaker $W$ function, and $a,b\in[0,1/2]$. Using the Whittaker differential equations that $W_{0,a}(x)$ and $W_{1,b}(x)$ solve by definition, the integral in question can be shown to effectively reduce to the following one $$\int W_{0,a}(x)\,W_{1,b}(x)\,\frac{dx}{x^2}\;\text{or}\; \int W_{0,a}(1/x)\,W_{1,b}(1/x)\,dx$$ which I'm also not sure how to find. Any suggestions?

Identities from MathematicalFunctionData can be useful when Mathematica can't seem to rewrite things in a form we want.

whit[k_, m_, z_] = Activate[MathematicalFunctionData[
WhittakerW, "AlternativeRepresentations"][[4]][k, m, z][[1, 2]]]

int = FullSimplify[whit[0, a, x] whit[1, b, x]]

Integrate[int/x, x]
(* large output of HypergeometricPFQs *)
• What a find. Many thanks. (+1) – bbgodfrey May 6 '16 at 0:13

Mathematica can't this integral to solve,but we can convert WhittakerW function to BesselI function.I'm use Maple to convert.

$$W_{0,a}(x)=\frac{1}{2} \sqrt{\pi } \sqrt{x} \csc (\pi (a+1)) \left(I_a\left(\frac{x}{2}\right)-I_{-a}\left(\frac{x}{2}\right)\right)$$ $$W_{1,b}(x)=\frac{1}{2} \sqrt{\pi } \sqrt{x} \csc (\pi b) \left(-\frac{1}{2} x I_{1-b}\left(\frac{x}{2}\right)+\left(b+\frac{x}{2}-\frac{1}{2}\right) I_{-b}\left(\frac{x}{2}\right)+\left(b-\frac{x}{2}+\frac{1}{2}\right) I_b\left(\frac{x}{2}\right)+\frac{1}{2} x I_{b+1}\left(\frac{x}{2}\right)\right)$$

whit1 = 1/2*Sqrt[Pi*x]/Sin[Pi*(1 + a)]*(BesselI[a, x/2] - BesselI[-a, x/2]);
whit2 = 1/2*Sqrt[Pi*x]/Sin[Pi*b]*(-BesselI[1 - b, x/2]*x/2 + BesselI[-b, x/2]*(b + x/2 - 1/2) + BesselI[1 + b, x/2]*x/2 + BesselI[b, x/2]*(b - x/2 + 1/2));
int = FullSimplify[whit1*whit2/x]

(* 1/4 (-x BesselI[1 - b, x/2] + (-1 + 2 b + x) BesselI[-b, x/2] + (1 + 2 b - x) BesselI[b, x/2] + x BesselI[1 + b, x/2]) BesselK[a, x/2] Csc[b \[Pi]] *)

integral = Integrate[int, x, Assumptions -> {a, b} \[Element] Reals]

(* 1/4 Csc[b \[Pi]] (-(1/2) \[Pi] Csc[
a \[Pi]] ((2^(-1 + 2 a + 2 b) x^(1 - a - b)
HypergeometricPFQ[{1/2 - a/2 - b/2, 1/2 - a/2 - b/2,
1 - a/2 - b/2}, {1 - a, 1 - b, 1 - a - b, 3/2 - a/2 - b/2},
x^2/4])/((1/2 - a/2 - b/2) Gamma[1 - a] Gamma[
1 - b]) - (2^(-1 - 2 a + 2 b) x^(1 + a - b)
HypergeometricPFQ[{1/2 + a/2 - b/2, 1/2 + a/2 - b/2,
1 + a/2 - b/2}, {1 + a, 1 - b, 1 + a - b, 3/2 + a/2 - b/2},
x^2/4])/((1/2 + a/2 - b/2) Gamma[1 + a] Gamma[1 - b])) +
b \[Pi] Csc[
a \[Pi]] ((2^(-1 + 2 a + 2 b) x^(1 - a - b)
HypergeometricPFQ[{1/2 - a/2 - b/2, 1/2 - a/2 - b/2,
1 - a/2 - b/2}, {1 - a, 1 - b, 1 - a - b, 3/2 - a/2 - b/2},
x^2/4])/((1/2 - a/2 - b/2) Gamma[1 - a] Gamma[
1 - b]) - (2^(-1 - 2 a + 2 b) x^(1 + a - b)
HypergeometricPFQ[{1/2 + a/2 - b/2, 1/2 + a/2 - b/2,
1 + a/2 - b/2}, {1 + a, 1 - b, 1 + a - b, 3/2 + a/2 - b/2},
x^2/4])/((1/2 + a/2 - b/2) Gamma[1 + a] Gamma[1 - b])) +
1/2 \[Pi] Csc[
a \[Pi]] ((2^(-1 + 2 a + 2 b) x^(2 - a - b)
HypergeometricPFQ[{1/2 - a/2 - b/2, 1 - a/2 - b/2,
1 - a/2 - b/2}, {1 - a, 1 - b, 1 - a - b, 2 - a/2 - b/2},
x^2/4])/((1 - a/2 - b/2) Gamma[1 - a] Gamma[
1 - b]) - (2^(-1 - 2 a + 2 b) x^(2 + a - b)
HypergeometricPFQ[{1/2 + a/2 - b/2, 1 + a/2 - b/2,
1 + a/2 - b/2}, {1 + a, 1 - b, 1 + a - b, 2 + a/2 - b/2},
x^2/4])/((1 + a/2 - b/2) Gamma[1 + a] Gamma[1 - b])) -
1/2 \[Pi] Csc[
a \[Pi]] ((2^(-3 + 2 a + 2 b) x^(3 - a - b)
HypergeometricPFQ[{1 - a/2 - b/2, 3/2 - a/2 - b/2,
3/2 - a/2 - b/2}, {1 - a, 2 - b, 2 - a - b,
5/2 - a/2 - b/2}, x^2/4])/((3/2 - a/2 - b/2) Gamma[
1 - a] Gamma[2 - b]) - (2^(-3 - 2 a + 2 b) x^(3 + a - b)
HypergeometricPFQ[{1 + a/2 - b/2, 3/2 + a/2 - b/2,
3/2 + a/2 - b/2}, {1 + a, 2 - b, 2 + a - b,
5/2 + a/2 - b/2}, x^2/4])/((3/2 + a/2 - b/2) Gamma[
1 + a] Gamma[2 - b])) +
1/2 \[Pi] Csc[
a \[Pi]] ((2^(-1 + 2 a - 2 b) x^(1 - a + b)
HypergeometricPFQ[{1/2 - a/2 + b/2, 1/2 - a/2 + b/2,
1 - a/2 + b/2}, {1 - a, 3/2 - a/2 + b/2, 1 + b, 1 - a + b},
x^2/4])/((1/2 - a/2 + b/2) Gamma[1 - a] Gamma[
1 + b]) - (2^(-1 - 2 a - 2 b) x^(1 + a + b)
HypergeometricPFQ[{1/2 + a/2 + b/2, 1/2 + a/2 + b/2,
1 + a/2 + b/2}, {1 + a, 3/2 + a/2 + b/2, 1 + b, 1 + a + b},
x^2/4])/((1/2 + a/2 + b/2) Gamma[1 + a] Gamma[1 + b])) +
b \[Pi] Csc[
a \[Pi]] ((2^(-1 + 2 a - 2 b) x^(1 - a + b)
HypergeometricPFQ[{1/2 - a/2 + b/2, 1/2 - a/2 + b/2,
1 - a/2 + b/2}, {1 - a, 3/2 - a/2 + b/2, 1 + b, 1 - a + b},
x^2/4])/((1/2 - a/2 + b/2) Gamma[1 - a] Gamma[
1 + b]) - (2^(-1 - 2 a - 2 b) x^(1 + a + b)
HypergeometricPFQ[{1/2 + a/2 + b/2, 1/2 + a/2 + b/2,
1 + a/2 + b/2}, {1 + a, 3/2 + a/2 + b/2, 1 + b, 1 + a + b},
x^2/4])/((1/2 + a/2 + b/2) Gamma[1 + a] Gamma[1 + b])) -
1/2 \[Pi] Csc[
a \[Pi]] ((2^(-1 + 2 a - 2 b) x^(2 - a + b)
HypergeometricPFQ[{1/2 - a/2 + b/2, 1 - a/2 + b/2,
1 - a/2 + b/2}, {1 - a, 2 - a/2 + b/2, 1 + b, 1 - a + b},
x^2/4])/((1 - a/2 + b/2) Gamma[1 - a] Gamma[
1 + b]) - (2^(-1 - 2 a - 2 b) x^(2 + a + b)
HypergeometricPFQ[{1/2 + a/2 + b/2, 1 + a/2 + b/2,
1 + a/2 + b/2}, {1 + a, 2 + a/2 + b/2, 1 + b, 1 + a + b},
x^2/4])/((1 + a/2 + b/2) Gamma[1 + a] Gamma[1 + b])) +
1/2 \[Pi] Csc[
a \[Pi]] ((2^(-3 + 2 a - 2 b) x^(3 - a + b)
HypergeometricPFQ[{1 - a/2 + b/2, 3/2 - a/2 + b/2,
3/2 - a/2 + b/2}, {1 - a, 5/2 - a/2 + b/2, 2 + b,
2 - a + b}, x^2/4])/((3/2 - a/2 + b/2) Gamma[1 - a] Gamma[
2 + b]) - (2^(-3 - 2 a - 2 b) x^(3 + a + b)
HypergeometricPFQ[{1 + a/2 + b/2, 3/2 + a/2 + b/2,
3/2 + a/2 + b/2}, {1 + a, 5/2 + a/2 + b/2, 2 + b,
2 + a + b}, x^2/4])/((3/2 + a/2 + b/2) Gamma[1 + a] Gamma[
2 + b]))) *)

Now we check numerical:

a = 0;
b = 1/2;
Integrate[WhittakerW[0, a, x]*WhittakerW[1, b, x]/x, x]
MeijerG[{{1}, {2}}, {{3/2, 3/2}, {0}}, x] /. x -> 1 // N

$0.452954$

Needs["NumericalCalculus`"]
NLimit[integral /. x -> 1 /. b -> 1/2, a -> 0, Method -> SequenceLimit] // Quiet

$0.452954$