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Abstract <p> In the first quadrant of the complex plane, a new representation of the countable-valued function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\operatorname{\mathrm{Arg}}\Gamma(z)\)</EquationSource> </InlineEquation> containing an improper integral of a special form is found. A formula for the principal argument is also discussed. This result, together with known properties of the gamma function, makes it possible to evaluate <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\arg\Gamma(z)\in(-\pi,\pi]\)</EquationSource> </InlineEquation> at points <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(z\)</EquationSource> </InlineEquation> of the other quadrants. Illustrative examples are analyzed. The obtained relations may be useful, e.g., in solving problems related to asymptotics of solutions to nonlinear differential equations of mathematical physics. </p>

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

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • A. B. Kostin,
  • V. B. Sherstyukov

摘要

Abstract

In the first quadrant of the complex plane, a new representation of the countable-valued function \(\operatorname{\mathrm{Arg}}\Gamma(z)\) containing an improper integral of a special form is found. A formula for the principal argument is also discussed. This result, together with known properties of the gamma function, makes it possible to evaluate \(\arg\Gamma(z)\in(-\pi,\pi]\) at points \(z\) of the other quadrants. Illustrative examples are analyzed. The obtained relations may be useful, e.g., in solving problems related to asymptotics of solutions to nonlinear differential equations of mathematical physics.