Bateman transform

In the <a href="/facts/Mathematics/pxTouaz4">mathematical</a> study of <a href="/facts/Partial_differential_equations/DyPsBlv7">partial differential equations</a>, the Bateman transform is a method for solving the <a href="/facts/Laplace_equation/8n1YkmFS">Laplace equation</a> in four dimensions and <a href="/facts/Wave_equation/J2ksD2Az">wave equation</a> in three by using a <a href="/facts/Line_integral/10s4B3Nn">line integral</a> of a <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic function</a> in three <a href="/facts/Complex_number/2w7mqI7e">complex variables</a>. It is named after the mathematician <a href="/facts/Harry_Bateman/9tRXIjI2">Harry Bateman</a>, who first published the result in (Bateman 1904).
The formula asserts that if ƒ is a holomorphic function of three complex variables, then

ϕ
        (
        w
        ,
        x
        ,
        y
        ,
        z
        )
        =
        
          ∮
          
            γ
          
        
        f
        
          
            (
          
        
        (
        w
        +
        i
        x
        )
        +
        (
        i
        y
        +
        z
        )
        ζ
        ,
        (
        i
        y
        −
        z
        )
        +
        (
        w
        −
        i
        x
        )
        ζ
        ,
        ζ
        
          
            )
          
        
        
        d
        ζ
      
    
    {\displaystyle \phi (w,x,y,z)=\oint _{\gamma }f{\big (}(w+ix)+(iy+z)\zeta ,(iy-z)+(w-ix)\zeta ,\zeta {\big )}\,d\zeta }

is a solution of the Laplace equation, which follows by differentiation under the integral. Furthermore, Bateman asserted that the most general solution of the Laplace equation arises in this way.

<ul><li><a href="/facts/Harry_Bateman/9tRXIjI2">Bateman, Harry</a> (1904), <a href="https://archive.today/20130415131037/http://plms.oxfordjournals.org/cgi/reprint/s2-1/1/451">"The solution of partial differential equations by means of definite integrals"</a>, <a href="/facts/Proceedings_of_the_London_Mathematical_Society/jdcf3vhX">Proceedings of the London Mathematical Society</a>, 1 (1): 451–458, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fplms%2Fs2-1.1.451">10.1112/plms/s2-1.1.451</a>, archived from <a href="http://plms.oxfordjournals.org/cgi/reprint/s2-1/1/451">the original</a> on 2013-04-15.</li>
<li><a href="/facts/Michael_Eastwood/XxVdPUEL">Eastwood, Michael</a> (2002), <a href="http://www.msri.org/ext/concepts/eastwood3.pdf">Bateman's formula</a> (PDF), <a href="/facts/Mathematical_Sciences_Research_Institute/Kvc4tGVl">MSRI</a>.</li></ul>

Bateman transform open-in-new

Bateman transform