- Source: Lommel function
The Lommel differential equation, named after Eugen von Lommel, is an inhomogeneous form of the Bessel differential equation:
z
2
d
2
y
d
z
2
+
z
d
y
d
z
+
(
z
2
−
ν
2
)
y
=
z
μ
+
1
.
{\displaystyle z^{2}{\frac {d^{2}y}{dz^{2}}}+z{\frac {dy}{dz}}+(z^{2}-\nu ^{2})y=z^{\mu +1}.}
Solutions are given by the Lommel functions sμ,ν(z) and Sμ,ν(z), introduced by Eugen von Lommel (1880),
s
μ
,
ν
(
z
)
=
π
2
[
Y
ν
(
z
)
∫
0
z
x
μ
J
ν
(
x
)
d
x
−
J
ν
(
z
)
∫
0
z
x
μ
Y
ν
(
x
)
d
x
]
,
{\displaystyle s_{\mu ,\nu }(z)={\frac {\pi }{2}}\left[Y_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }J_{\nu }(x)\,dx-J_{\nu }(z)\!\int _{0}^{z}\!\!x^{\mu }Y_{\nu }(x)\,dx\right],}
S
μ
,
ν
(
z
)
=
s
μ
,
ν
(
z
)
+
2
μ
−
1
Γ
(
μ
+
ν
+
1
2
)
Γ
(
μ
−
ν
+
1
2
)
(
sin
[
(
μ
−
ν
)
π
2
]
J
ν
(
z
)
−
cos
[
(
μ
−
ν
)
π
2
]
Y
ν
(
z
)
)
,
{\displaystyle S_{\mu ,\nu }(z)=s_{\mu ,\nu }(z)+2^{\mu -1}\Gamma \left({\frac {\mu +\nu +1}{2}}\right)\Gamma \left({\frac {\mu -\nu +1}{2}}\right)\left(\sin \left[(\mu -\nu ){\frac {\pi }{2}}\right]J_{\nu }(z)-\cos \left[(\mu -\nu ){\frac {\pi }{2}}\right]Y_{\nu }(z)\right),}
where Jν(z) is a Bessel function of the first kind and Yν(z) a Bessel function of the second kind.
The s function can also be written as
s
μ
,
ν
(
z
)
=
z
μ
+
1
(
μ
−
ν
+
1
)
(
μ
+
ν
+
1
)
1
F
2
(
1
;
μ
2
−
ν
2
+
3
2
,
μ
2
+
ν
2
+
3
2
;
−
z
2
4
)
,
{\displaystyle s_{\mu ,\nu }(z)={\frac {z^{\mu +1}}{(\mu -\nu +1)(\mu +\nu +1)}}{}_{1}F_{2}(1;{\frac {\mu }{2}}-{\frac {\nu }{2}}+{\frac {3}{2}},{\frac {\mu }{2}}+{\frac {\nu }{2}}+{\frac {3}{2}};-{\frac {z^{2}}{4}}),}
where pFq is a generalized hypergeometric function.
See also
Anger function
Lommel polynomial
Struve function
Weber function
References
Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Higher transcendental functions. Vol II (PDF), McGraw-Hill Book Company, Inc., New York-Toronto-London, MR 0058756
Lommel, E. (1875), "Ueber eine mit den Bessel'schen Functionen verwandte Function", Math. Ann., 9 (3): 425–444, doi:10.1007/BF01443342
Lommel, E. (1880), "Zur Theorie der Bessel'schen Funktionen IV", Math. Ann., 16 (2): 183–208, doi:10.1007/BF01446386
Paris, R. B. (2010), "Lommel 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.
Solomentsev, E.D. (2001) [1994], "Lommel function", Encyclopedia of Mathematics, EMS Press
External links
Weisstein, Eric W. "Lommel Differential Equation." From MathWorld—A Wolfram Web Resource.
Weisstein, Eric W. "Lommel Function." From MathWorld—A Wolfram Web Resource.
Kata Kunci Pencarian:
- Isotop roentgenium
- Roentgenium
- Isotop meitnerium
- Lommel function
- Bessel function
- Pim van Lommel
- Lommel polynomial
- Lommel (disambiguation)
- Eugen von Lommel
- Anger function
- List of eponyms of special functions
- Generalized hypergeometric function
- Fresnel diffraction
