In mathematics , the Struve functions H α x )y (x )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)}}}
Graph of
H
n
(
x
)
{\displaystyle \mathrm {H} _{n}(x)}
n
∈
[
0
,
1
,
2
,
3
,
4
,
5
]
{\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
K
α
(
x
)
{\displaystyle \mathbf {K} _{\alpha }(x)}
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 )−ie −iαπ / 2 H α ix ) , are solutions y (x )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)}
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
Struve functions, denoted as H α z )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 )
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 )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 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 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] [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 . 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 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 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 . 
_with_n=2_in_the_complex_plane_from_-2-2i_to_2+2i_with_colors_created_with_Mathematica_13.1_function_ComplexPlot3D.svg)
_with_n=2_in_the_complex_plane_from_-2-2i_to_2+2i_with_colors_created_with_Mathematica_13.1_function_ComplexPlot3D.svg)
![{\displaystyle n\in [0,1,2,3,4,5]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/bff616f773f808ad1bf4d829aeb4565f76fd3446)
