Parabolic cylinder function

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In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabolic cylindrical coordinates. The above equation may be brought into two distinct forms (A) and (B) by completing the square and rescaling z, called H. F. Weber's equations: and If f ( a , z ) {\displaystyle f(a,z)} is a solution, then so are f ( a , − z ) , f ( − a , i z ) and f ( − a , − i z ) . {\displaystyle f(a,-z),f(-a,iz){\text{ and }}f(-a,-iz).} If f ( a , z ) {\displaystyle f(a,z)\,} is a solution of equation (A), then f ( − i a , z e ( 1 / 4 ) π i ) {\displaystyle f(-ia,ze^{(1/4)\pi i})} is a solution of (B), and, by symmetry, f ( − i a , − z e ( 1 / 4 ) π i ) , f ( i a , − z e − ( 1 / 4 ) π i ) and f ( i a , z e − ( 1 / 4 ) π i ) {\displaystyle f(-ia,-ze^{(1/4)\pi i}),f(ia,-ze^{-(1/4)\pi i}){\text{ and }}f(ia,ze^{-(1/4)\pi i})} are also solutions of (B).