In mathematics, the Anger function, introduced by C. T. Anger (1855), is a function defined as
J
ν
(
z
)
=
1
π
∫
0
π
cos
(
ν
θ
−
z
sin
θ
)
d
θ
{\displaystyle \mathbf {J} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\nu \theta -z\sin \theta )\,d\theta }
with complex parameter
ν
{\displaystyle \nu }
and complex variable
z
{\displaystyle {\textit {z}}}
. It is closely related to the Bessel functions.
The Weber function (also known as Lommel–Weber function), introduced by H. F. Weber (1879), is a closely related function defined by
E
ν
(
z
)
=
1
π
∫
0
π
sin
(
ν
θ
−
z
sin
θ
)
d
θ
{\displaystyle \mathbf {E} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\sin(\nu \theta -z\sin \theta )\,d\theta }
and is closely related to Bessel functions of the second kind.
Contents
Relation between Weber and Anger functions
The Anger and Weber functions are related by
sin
(
π
ν
)
J
ν
(
z
)
=
cos
(
π
ν
)
E
ν
(
z
)
−
E
−
ν
(
z
)
,
−
sin
(
Power series expansion
The Anger function has the power series expansion
J
ν
(
z
)
=
cos
π
ν
2
∑
k
=
0
∞
(
−
1
)
k
z
2
k
4
k
Γ
(
k
+
ν
2
+
1
)
Γ
Differential equations
The Anger and Weber functions are solutions of inhomogeneous forms of Bessel's equation
z
2
y
′
′
+
z
y
′
+
(
z
2
−
ν
2
)
y
=
0.
{\displaystyle z^{2}y^{\prime \prime }+zy^{\prime }+(z^{2}-\nu ^{2})y=0.}
More precisely, the Anger functions satisfy the equation
z
2
y
′
′
+
z
y
′
+
Recurrence relations
The Anger function satisfies this inhomogeneous form of recurrence relation
z
J
ν
−
1
(
z
)
+
z
J
ν
+
1
(
z
)
=
2
ν
J
ν
(
z
)
−
2
sin
π
ν
π
.
{\displaystyle z\mathbf {J} _{\nu -1}(z)+z\mathbf {J} _{\nu +1}(z)=2\nu \mathbf {J} _{\nu }(z)-{\frac {2\sin \pi \nu }{\pi }}.}
Delay differential equations
The Anger and Weber functions satisfy these homogeneous forms of delay differential equations
J
ν
−
1
(
z
)
−
J
ν
+
1
(
z
)
=
2
∂
∂
z
J
ν
(
z
)
,
{\displaystyle \mathbf {J} _{\nu -1}(z)-\mathbf {J} _{\nu +1}(z)=2{\dfrac {\partial }{\partial z}}\mathbf {J} _{\nu }(z),}
E
ν
−
1
(