The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general. Other fractional arguments can be approximated through efficient infinite products, infinite series, and recurrence relations.
It is unknown whether these constants are transcendental in general, but Γ(1/3) and Γ(1/4) were shown to be transcendental by G. V. Chudnovsky. Γ(1/4) /4√π has also long been known to be transcendental, and Yuri Nesterenko proved in 1996 that Γ(1/4), π, and eπ are algebraically independent.
Because of the Euler Reflection Formula, and the fact that , we have an expression for the modulus squared of the gamma function evaluated on the imaginary axis:
The above integral therefore relates to the phase of .
The gamma function with other complex arguments returns
Other constants
The gamma function has a local minimum on the positive real axis
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In mathematics, the modular lambda function λ(τ) is a highly symmetric Holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X(2). Over any point τ, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve , where the map is defined as the quotient by the [−1] involution.
↑ Borwein, J. M.; Zucker, I. J. (1992). "Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind". IMA Journal of Numerical Analysis. 12 (4): 519–526. doi:10.1093/imanum/12.4.519. MR1186733.
↑ Mező, István (2013), "Duplication formulae involving Jacobi theta functions and Gosper's q-trigonometric functions", Proceedings of the American Mathematical Society, 141 (7): 2401–2410, doi:10.1090/s0002-9939-2013-11576-5
↑ Johansson, F. (2023). Arbitrary-precision computation of the gamma function. Maple Transactions, 3(1). doi:10.5206/mt.v3i1.14591
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