Mathquery

Confluent Hypergeometric Function :       Let us put x = z/β or z= βx in the hypergeometric equation . Then the equation assumes the form   z(1- z/β)d²y/dz²  +                 [γ-(1+α+β)z/β]dy/dz -αy = 0 There are three regular singularities , one at z= 0 , another given by z= β , and also at z=∞.     When β-->∞ , we get the equation  z = d²y/dz² + (γ-z)dy/dz -αy = 0   .......(1)    The equation is known as the Confluent Hypergeometric Equation following to confluence of two singularities β,∞ when      β-->∞  .   Again putting x= z/β in the hypergeometric series we obtain  1+ α.βz/1.γβ + α(α+1).β(β+1)z²/1.2.γ(γ+1)β²+..   = 1+ αz/1.γ + α(α+1)(1+ 1/β)z²/1.2.γ(γ+1)+... When β-->∞ this series reduces to   1+αz/1.γ  + α(α+1)z²/2!γ(γ+1) + .... which can be put as        ∞       Σ (α)ₖ zᵏ/k! (γ)ₖ   or simply F(α;γ;z) .   ...(2)    k=0 This function is called Confluent Hypergeometric Function . Note :