Graph of H n ( x ) {\displaystyle \mathrm {H} _{n}(x)} for n ∈ [ 0 , 1 , 2 , 3 , 4 , 5 ] {\displaystyle n\in [0,1,2,3,4,5]} 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)} .
The modified Struve functions L α (x ) are equal to −ie −iαπ / 2 H α (ix ) , 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 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)} .
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 [ edit ] 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}.} 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 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 Yα (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}}} Relation to other functions [ edit ] 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 1 F 2 :
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]
^ K. Buchanan, C. Flores, S. Wheeland, J. Jensen, D. Grayson and G. Huff, "Transmit beamforming for radar applications using circularly tapered random arrays," 2017 IEEE Radar Conference (RadarConf), 2017, pp. 0112-0117, doi: 10.1109/RADAR.2017.7944181 ^ B. U. Felderhof, "Effect of the wall on the velocity autocorrelation function and long-time tail of Brownian motion." The Journal of Physical Chemistry B 109.45, 2005, pp. 21406-21412 R. M. Aarts and Augustus J. E. M. Janssen (2003). "Approximation of the Struve function H 1 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 .