Struveove funkcije
H
α
(
x
)
{\displaystyle \mathbf {H} _{\alpha }(x)}
Beselove diferencijalne jednačine :
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}}+(x^{2}-\alpha ^{2})y={\frac {4{(x/2)}^{\alpha +1}}{{\sqrt {\pi }}\Gamma (\alpha +{\frac {1}{2}})}}}
Funkcije su nazvane prema ruskom astronomu Hermanu Struveu , koji ih je otkrio 1882 . da bi rešio izvesne probleme luminoziteta u astronomiji . Kompleksni broj α predstavlja red Struveove funkcije. Modifikovana Struveova funkcija
L
α
(
x
)
{\displaystyle \mathbf {L} _{\alpha }(x)}
−
i
e
−
i
α
π
/
2
H
α
(
i
x
)
{\displaystyle -ie^{-i\alpha \pi /2}\mathbf {H} _{\alpha }(ix)}
Homogeno rešenje Beselove jednačine su Beselove funkcije , a rešenje nehomogene gorenavedene Beselove jednačine je Struveova funkcija, koja razvijena u red ima sledeći oblik:
H
α
(
x
)
=
∑
m
=
0
∞
(
−
1
)
m
Γ
(
m
+
3
2
)
Γ
(
m
+
α
+
3
2
)
(
x
2
)
2
m
+
α
+
1
{\displaystyle \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}}
gde je
Γ
(
z
)
{\displaystyle \Gamma (z)}
gama funkcija .
Modifikovana Struveova funkcija
L
ν
(
z
)
{\displaystyle \mathbf {L} _{\nu }(z)}
L
ν
(
z
)
=
(
z
2
)
ν
+
1
∑
k
=
0
∞
1
Γ
(
3
2
+
k
)
Γ
(
3
2
+
k
+
ν
)
(
z
2
)
2
k
{\displaystyle \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}}
Integralni oblik Struveove funkcije je:
H
α
(
x
)
=
2
(
x
/
2
)
α
π
Γ
(
α
+
1
2
)
∫
0
π
/
2
sin
(
x
cos
τ
)
sin
2
α
(
τ
)
d
τ
.
{\displaystyle \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 .}
uredi
Struveove funkcije zadovoljavaju sledeće rekurzivne relacije:
H
α
−
1
(
x
)
+
H
α
+
1
(
x
)
=
2
α
x
H
α
(
x
)
+
(
x
/
2
)
α
π
Γ
(
α
+
3
2
)
{\displaystyle \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}})}}}
H
α
−
1
(
x
)
−
H
α
+
1
(
x
)
=
2
d
H
α
d
x
−
(
x
/
2
)
α
π
Γ
(
α
+
3
2
)
.
{\displaystyle \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}})}}.}
uredi
Struveove funkcije celobrojnoga reda mogu da se prikažu pomoću Veberovih funkcija E n n nenegativni celi broj onda je:
E
n
(
z
)
=
1
π
∑
k
=
0
[
n
−
1
2
]
Γ
(
k
+
1
/
2
)
(
z
/
2
)
n
−
2
k
−
1
Γ
(
n
−
1
/
2
−
k
)
H
n
{\displaystyle \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}}
E
−
n
(
z
)
=
(
−
1
)
n
+
1
π
∑
k
=
0
[
n
−
1
2
]
Γ
(
n
−
k
−
1
/
2
)
(
z
/
2
)
−
n
+
2
k
+
1
Γ
(
k
+
3
/
2
)
H
−
n
.
{\displaystyle \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}.}
Struveove funkcije reda n +1/2 (n je celi broj) mogu da se izraze preko Beselovih funkcija :
H
−
n
−
1
/
2
(
z
)
=
(
−
1
)
n
J
n
+
1
/
2
(
z
)
{\displaystyle \mathbf {H} _{-n-1/2}(z)=(-1)^{n}J_{n+1/2}(z)}
Struveove funkcije mogu da se predstave i pomoću poopštene hipergeometrijske funkcije :
H
α
(
z
)
=
(
z
/
2
)
α
+
1
/
2
2
π
Γ
(
α
+
3
/
2
)
1
F
2
(
1
,
3
/
2
,
α
+
3
/
2
,
−
z
2
/
4
)
.
{\displaystyle \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).}
Za veliko x dobija se asimptotski oblik:
H
α
(
x
)
−
Y
α
(
x
)
→
1
π
Γ
(
α
+
1
2
)
(
x
2
)
α
−
1
+
O
(
(
x
/
2
)
α
−
3
)
{\displaystyle \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)}
gde je
Y
α
(
x
)
{\displaystyle Y_{\alpha }(x)}
Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers , McGraw-Hill.
Abramowitz, Milton; Stegun, Irene A., eds. (1965), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, ISBN 978-0486612720
Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. ISBN 0-07-043316-X
Struveove funkcije 
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8961b7a2742207fc052ad365047604f6f8aad312)
![{\displaystyle \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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69d579072e41fc2192afef7163a14dc50e5c7c38)
