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Definition

Tricomi U for positive real arguments

A convergent Laplace integral defining a confluent hypergeometric function on a positive real domain.

For a>0a>0, b∈Rb\in\mathbb R, and z>0z>0, the Tricomi confluent hypergeometric function is represented by

U(a,b,z)=1Γ(a)∫0∞e−ztta−1(1+t)b−a−1 dt.U(a,b,z)=\frac1{\Gamma(a)}\int_0^\infty e^{-zt}t^{a-1}(1+t)^{b-a-1}\,dt.

The fixes its normalization. The positive integrand is integrable at zero and infinity in this domain.

Differential equation

Differentiating under the integral and integrating the derivative of e−ztta(1+t)b−ae^{-zt}t^a(1+t)^{b-a} shows

zUzz+(b−z)Uz−aU=0.zU_{zz}+(b-z)U_z-aU=0.

Both boundary terms vanish. This real integral domain suffices for many positive-profile calculations; continuation to complex arguments entails additional branch conventions.

References