Convergence of Magnus integral addition theorems for confluent hypergeometric functions

Area of Science:

• Mathematical Physics
• Special Functions
• Asymptotic Analysis

Background:

• Magnus presented an addition theorem for the confluent hypergeometric function U in 1946.
• The theorem expresses U(x+y) as an integral of U(x)U(y).
• The original theorem's convergence domain was not precisely determined.

Purpose of the Study:

• To determine the precise domain of convergence for Magnus' addition theorem.
• To derive integral addition theorems with exact convergence domains for related special functions.

Main Methods:

• Utilizing recently obtained asymptotics for the confluent hypergeometric function U with a large complex first parameter.
• Applying well-known specializations of the confluent hypergeometric function U.

Main Results:

• A precise domain of convergence for Magnus' addition theorem for U was determined.
• Integral addition theorems with precise convergence domains were obtained for modified parabolic cylinder functions.
• Integral addition theorems with precise convergence domains were obtained for Hankel, Macdonald, and Bessel functions (orders 0 and 1).

Conclusions:

• The study successfully established the convergence domain for a key addition theorem.
• The methodology provides a framework for analyzing convergence properties of related special function identities.
• This work offers precise convergence conditions crucial for numerical and theoretical applications of these functions.