Anger function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Anger function, introduced by <a href="/facts/Carl_Theodor_Anger/AJ7Rl5Kw">C. T. Anger</a> (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 <a href="/facts/Bessel_function/hSyvFPqC">Bessel functions</a>.
The Weber function (also known as <a href="/facts/Eugen_von_Lommel/sG1hvxFB">Lommel</a>–Weber function), introduced by <a href="/facts/Heinrich_Friedrich_Weber/GT7ylX6X">H. F. Weber</a> (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 <a href="/facts/Bessel_function/hSyvFPqC">Bessel functions</a> of the second kind.

Anger function open-in-new

Anger function