Struve function

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

x 2 d 2 y d x 2 + x d y d x + ( x 2 − α 2 ) y = 4 ( x 2 ) α + 1 π Γ ( α + 1 2 ) {\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)}}}

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 K α ( x ) {\displaystyle \mathbf {K} _{\alpha }(x)} as K α ( x ) = H α ( x ) − Y α ( x ) {\displaystyle \mathbf {K} _{\alpha }(x)=\mathbf {H} _{\alpha }(x)-Y_{\alpha }(x)} , where Y α ( x ) {\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:

x 2 d 2 y d x 2 + x d y d x − ( x 2 + α 2 ) y = 4 ( x 2 ) α + 1 π Γ ( α + 1 2 ) {\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 M α ( x ) {\displaystyle \mathbf {M} _{\alpha }(x)} as M α ( x ) = L α ( x ) − I α ( x ) {\displaystyle \mathbf {M} _{\alpha }(x)=\mathbf {L} _{\alpha }(x)-I_{\alpha }(x)} , where I α ( x ) {\displaystyle I_{\alpha }(x)} is the modified Bessel function.

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.

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

H α ( z ) = ∑ m = 0 ∞ ( − 1 ) m Γ ( m + 3 2 ) Γ ( m + α + 3 2 ) ( z 2 ) 2 m + α + 1 , {\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

L α ( z ) = ∑ m = 0 ∞ 1 Γ ( m + 3 2 ) Γ ( m + α + 3 2 ) ( z 2 ) 2 m + α + 1 . {\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}.}

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:

H α ( x ) = 2 ( x 2 ) α π Γ ( α + 1 2 ) ∫ 0 1 ( 1 − t 2 ) α − 1 2 sin ⁡ x t   d t = 2 ( x 2 ) α π Γ ( α + 1 2 ) ∫ 0 π 2 sin ⁡ ( x cos ⁡ τ ) sin 2 α ⁡ τ   d τ = 2 ( x 2 ) α π Γ ( α + 1 2 ) ∫ 0 π 2 sin ⁡ ( x sin ⁡ τ ) cos 2 α ⁡ τ   d τ {\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 }

K α ( x ) = 2 ( x 2 ) α π Γ ( α + 1 2 ) ∫ 0 ∞ ( 1 + t 2 ) α − 1 2 e − x t   d t = 2 ( x 2 ) α π Γ ( α + 1 2 ) ∫ 0 ∞ e − x sinh ⁡ τ cosh 2 α ⁡ τ   d τ {\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 }

L α ( x ) = 2 ( x 2 ) α π Γ ( α + 1 2 ) ∫ 0 1 ( 1 − t 2 ) α − 1 2 sinh ⁡ x t   d t = 2 ( x 2 ) α π Γ ( α + 1 2 ) ∫ 0 π 2 sinh ⁡ ( x cos ⁡ τ ) sin 2 α ⁡ τ   d τ = 2 ( x 2 ) α π Γ ( α + 1 2 ) ∫ 0 π 2 sinh ⁡ ( x sin ⁡ τ ) cos 2 α ⁡ τ   d τ {\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 }

M α ( x ) = − 2 ( x 2 ) α π Γ ( α + 1 2 ) ∫ 0 1 ( 1 − t 2 ) α − 1 2 e − x t   d t = − 2 ( x 2 ) α π Γ ( α + 1 2 ) ∫ 0 π 2 e − x cos ⁡ τ sin 2 α ⁡ τ   d τ = − 2 ( x 2 ) α π Γ ( α + 1 2 ) ∫ 0 π 2 e − x sin ⁡ τ cos 2 α ⁡ τ   d τ {\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 }

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

For large x, one obtains:

H α ( x ) − Y α ( x ) = ( x 2 ) α − 1 π Γ ( α + 1 2 ) + O ( ( x 2 ) α − 3 ) , {\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.

The Struve functions satisfy the following recurrence relations:

H α − 1 ( x ) + H α + 1 ( x ) = 2 α x H α ( x ) + ( x 2 ) α π Γ ( α + 3 2 ) , H α − 1 ( x ) − H α + 1 ( x ) = 2 d d x ( H α ( x ) ) − ( x 2 ) α π Γ ( α + 3 2 ) . {\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}}}

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

E n ( z ) = 1 π ∑ k = 0 ⌊ n − 1 2 ⌋ Γ ( k + 1 2 ) ( z 2 ) n − 2 k − 1 Γ ( n − k + 1 2 ) − H n ( z ) , E − n ( z ) = ( − 1 ) n + 1 π ∑ k = 0 ⌈ n − 3 2 ⌉ Γ ( n − k − 1 2 ) ( z 2 ) − n + 2 k + 1 Γ ( k + 3 2 ) − H − n ( z ) . {\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

H − n − 1 2 ( z ) = ( − 1 ) n J n + 1 2 ( z ) , {\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:

H α ( z ) = z α + 1 2 α π Γ ( α + 3 2 ) 1 F 2 ( 1 ; 3 2 , α + 3 2 ; − z 2 4 ) . {\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).}

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]

• R. M. Aarts and Augustus J. E. M. Janssen (2003). "Approximation of the Struve function H₁ occurring in impedance calculations". J. Acoust. Soc. Am. 113 (5): 2635–2637. Bibcode:2003ASAJ..113.2635A. doi:10.1121/1.1564019. PMID 12765381.
• R. M. Aarts and Augustus J. E. M. Janssen (2016). "Efficient approximation of the Struve functions H_n occurring in the calculation of sound radiation quantities". J. Acoust. Soc. Am. 140 (6): 4154–4160. Bibcode:2016ASAJ..140.4154A. doi:10.1121/1.4968792. PMID 28040027.
• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 12". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 496. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
• Ivanov, A. B. (2001) [1994], "Struve function", Encyclopedia of Mathematics, EMS Press
• Paris, R. B. (2010), "Struve function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Struve, H. (1882). "Beitrag zur Theorie der Diffraction an Fernröhren". Annalen der Physik und Chemie. 17 (13): 1008–1016. Bibcode:1882AnP...253.1008S. doi:10.1002/andp.18822531319.

• Struve functions at the Wolfram functions site.