Struve function

In mathematics, the Struve functions H_α(x), are solutions y(x) of the non-homogeneous Bessel's differential equation:

${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y={\frac {4\left({\frac {x}{2}}\right)^{\alpha +1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}}$

Graph of ${\displaystyle \mathrm {H} _{n}(x)}$ for ${\displaystyle n\in [0,1,2,3,4,5]}$

introduced by Hermann Struve (1882). The complex number α is the order of the Struve function, and is often an integer.

And further defined its second-kind version ${\displaystyle \mathbf {K} _{\alpha }(x)}$ as ${\displaystyle \mathbf {K} _{\alpha }(x)=\mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)}$, where ${\displaystyle Y_{\alpha }(x)}$ is the Neumann function.

The modified Struve functions L_α(x) are equal to −ie^{−iαπ / 2}H_α(ix) and are solutions y(x) of the non-homogeneous Bessel's differential equation:

Plot of the Struve function H n(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

${\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}-\left(x^{2}+\alpha ^{2}\right)y={\frac {4\left({\frac {x}{2}}\right)^{\alpha +1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}}$

And further defined its second-kind version ${\displaystyle \mathbf {M} _{\alpha }(x)}$ as ${\displaystyle \mathbf {M} _{\alpha }(x)=\mathbf {L} _{\alpha }(x)-I_{\alpha }(x)}$, where ${\displaystyle I_{\alpha }(x)}$ is the modified Bessel function.

Definitions

Since this is a non-homogeneous equation, solutions can be constructed from a single particular solution by adding the solutions of the homogeneous problem. In this case, the homogeneous solutions are the Bessel functions, and the particular solution may be chosen as the corresponding Struve function.

Power series expansion

Struve functions, denoted as H_α(z) have the power series form

${\displaystyle \mathbf {H} _{\alpha }(z)=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{\Gamma \left(m+{\frac {3}{2}}\right)\Gamma \left(m+\alpha +{\frac {3}{2}}\right)}}\left({\frac {z}{2}}\right)^{2m+\alpha +1},}$

where Γ(z) is the gamma function.

The modified Struve functions, denoted L_α(z), have the following power series form

${\displaystyle \mathbf {L} _{\alpha }(z)=\sum _{m=0}^{\infty }{\frac {1}{\Gamma \left(m+{\frac {3}{2}}\right)\Gamma \left(m+\alpha +{\frac {3}{2}}\right)}}\left({\frac {z}{2}}\right)^{2m+\alpha +1}.}$

Plot of the modified Struve function L n(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

Integral form

Another definition of the Struve function, for values of α satisfying Re(α) > − ⁠1/2⁠, is possible expressing in term of the Poisson's integral representation:

$${\displaystyle \mathbf {H} _{\alpha }(x)={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{1}(1-t^{2})^{\alpha -{\frac {1}{2}}}\sin xt~dt={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}\sin(x\cos \tau )\sin ^{2\alpha }\tau ~d\tau ={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}\sin(x\sin \tau )\cos ^{2\alpha }\tau ~d\tau }$$

$${\displaystyle \mathbf {K} _{\alpha }(x)={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\infty }(1+t^{2})^{\alpha -{\frac {1}{2}}}e^{-xt}~dt={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\infty }e^{-x\sinh \tau }\cosh ^{2\alpha }\tau ~d\tau }$$

$${\displaystyle \mathbf {L} _{\alpha }(x)={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{1}(1-t^{2})^{\alpha -{\frac {1}{2}}}\sinh xt~dt={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}\sinh(x\cos \tau )\sin ^{2\alpha }\tau ~d\tau ={\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}\sinh(x\sin \tau )\cos ^{2\alpha }\tau ~d\tau }$$

$${\displaystyle \mathbf {M} _{\alpha }(x)=-{\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{1}(1-t^{2})^{\alpha -{\frac {1}{2}}}e^{-xt}~dt=-{\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}e^{-x\cos \tau }\sin ^{2\alpha }\tau ~d\tau =-{\frac {2\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\frac {\pi }{2}}e^{-x\sin \tau }\cos ^{2\alpha }\tau ~d\tau }$$

Asymptotic forms

For small x, the power series expansion is given above.

For large x, one obtains:

${\displaystyle \mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)={\frac {\left({\frac {x}{2}}\right)^{\alpha -1}}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}+O\left(\left({\tfrac {x}{2}}\right)^{\alpha -3}\right),}$

where Y_α(x) is the Neumann function.

Properties

The Struve functions satisfy the following recurrence relations:

${\displaystyle {\begin{aligned}\mathbf {H} _{\alpha -1}(x)+\mathbf {H} _{\alpha +1}(x)&={\frac {2\alpha }{x}}\mathbf {H} _{\alpha }(x)+{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}},\\\mathbf {H} _{\alpha -1}(x)-\mathbf {H} _{\alpha +1}(x)&=2{\frac {d}{dx}}\left(\mathbf {H} _{\alpha }(x)\right)-{\frac {\left({\frac {x}{2}}\right)^{\alpha }}{{\sqrt {\pi }}\Gamma \left(\alpha +{\frac {3}{2}}\right)}}.\end{aligned}}}$

Relation to other functions

Struve functions of integer order can be expressed in terms of Weber functions E_n and vice versa: if n is a non-negative integer then

${\displaystyle {\begin{aligned}\mathbf {E} _{n}(z)&={\frac {1}{\pi }}\sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }{\frac {\Gamma \left(k+{\frac {1}{2}}\right)\left({\frac {z}{2}}\right)^{n-2k-1}}{\Gamma \left(n-k+{\frac {1}{2}}\right)}}-\mathbf {H} _{n}(z),\\\mathbf {E} _{-n}(z)&={\frac {(-1)^{n+1}}{\pi }}\sum _{k=0}^{\left\lceil {\frac {n-3}{2}}\right\rceil }{\frac {\Gamma (n-k-{\frac {1}{2}})\left({\frac {z}{2}}\right)^{-n+2k+1}}{\Gamma \left(k+{\frac {3}{2}}\right)}}-\mathbf {H} _{-n}(z).\end{aligned}}}$

Struve functions of order n + ⁠1/2⁠ where n is an integer can be expressed in terms of elementary functions. In particular if n is a non-negative integer then

${\displaystyle \mathbf {H} _{-n-{\frac {1}{2}}}(z)=(-1)^{n}J_{n+{\frac {1}{2}}}(z),}$

where the right hand side is a spherical Bessel function.

Struve functions (of any order) can be expressed in terms of the generalized hypergeometric function ₁F₂:

${\displaystyle \mathbf {H} _{\alpha }(z)={\frac {z^{\alpha +1}}{2^{\alpha }{\sqrt {\pi }}\Gamma \left(\alpha +{\tfrac {3}{2}}\right)}}{}_{1}F_{2}\left(1;{\tfrac {3}{2}},\alpha +{\tfrac {3}{2}};-{\tfrac {z^{2}}{4}}\right).}$

Applications

The Struve and Weber functions were shown to have an application to beamforming in.,^[1] and in describing the effect of confining interface on Brownian motion of colloidal particles at low Reynolds numbers.^[2]