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In mathematics, Struve functions $\mathbf {H} _{\alpha }(x)$, are solutions y(x) of the non-homogenous Bessel's differential equation:

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

introduced by Template:Harvs. The complex number α is the order of the Struve function, and is often an integer. The modified Struve functions $\mathbf {L} _{\alpha }(x)$ are equal to $-ie^{-i\alpha \pi /2}\mathbf {H} _{\alpha }(ix)$.

Definitions

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

Power series expansion

Struve functions, denoted as $\mathbf {H} _{\alpha }(x)$ have the following power series form

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

where $\Gamma (z)$ is the gamma function.

The modified Struve function, denoted as $\mathbf {L} _{\nu }(z)$ have the following power series form

$\mathbf {L} _{\nu }(z)={\left({\frac {z}{2}}\right)}^{\nu +1}\sum _{k=0}^{\infty }{\frac {1}{\Gamma ({\frac {3}{2}}+k)\Gamma ({\frac {3}{2}}+k+\nu )}}{\left({\frac {z}{2}}\right)}^{2k}$

Integral form

Another definition of the Struve function, for values of α satisfying $\operatorname {Re} \{\alpha \}>-1/2$, is possible using an integral representation:

$\mathbf {H} _{\alpha }(x)={\frac {2{(x/2)}^{\alpha }}{{\sqrt {\pi }}\Gamma (\alpha +{\frac {1}{2}})}}\int _{0}^{\pi /2}\sin(x\cos \tau )\sin ^{2\alpha }(\tau )d\tau .$

Asymptotic forms

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

For large x, one obtains:

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

where $Y_{\alpha }(x)$ is the Neumann function.

Properties

The Struve functions satisfy the following recurrence relations:

$\mathbf {H} _{\alpha -1}(x)+\mathbf {H} _{\alpha +1}(x)={\frac {2\alpha }{x}}\mathbf {H} _{\alpha }(x)+{\frac {{(x/2)}^{\alpha }}{{\sqrt {\pi }}\Gamma (\alpha +{\frac {3}{2}})}}$

$\mathbf {H} _{\alpha -1}(x)-\mathbf {H} _{\alpha +1}(x)=2{\frac {\mathrm {d} \mathbf {H} _{\alpha }}{\mathrm {d} x}}-{\frac {{(x/2)}^{\alpha }}{{\sqrt {\pi }}\Gamma (\alpha +{\frac {3}{2}})}}.$

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

$\mathbf {E} _{n}(z)={\frac {1}{\pi }}\sum _{k=0}^{[{\frac {n-1}{2}}]}{\frac {\Gamma (k+1/2)(z/2)^{n-2k-1}}{\Gamma (n-1/2-k)}}\mathbf {H} _{n}$

$\mathbf {E} _{-n}(z)={\frac {(-1)^{n+1}}{\pi }}\sum _{k=0}^{[{\frac {n-1}{2}}]}{\frac {\Gamma (n-k-1/2)(z/2)^{-n+2k+1}}{\Gamma (k+3/2)}}\mathbf {H} _{-n}.$

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

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

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 (which is not the Gauss hypergeometric function ₂F₁) :

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

References

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External links

• Struve functions at the Wolfram functions site.