An Analytical Formula for the Argument of the Gamma Function as a Complex Quantity

Key Points

• This research aims to present a new analytical formula for evaluating the argument of the gamma function in the complex plane.
• Developed a new representation of the argument function in the first quadrant of the complex plane.
• Derived a formula for the principal argument involving an improper integral.
• Explored how known properties of the gamma function can extend evaluations to other quadrants.
• Established a countable representation for the argument of the gamma function in the first quadrant.
• Demonstrated the ability to evaluate the argument at points in all quadrants.
• Provided illustrative examples showcasing practical applications to nonlinear differential equations.