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Jackson fall 2023 notes 14 the gamma function the gamma function appears in many expressions, including bessel functions, etc

It generalizes the factorial function n It appears in the method of steepest descent (a method for obtaining the asymptotic expansion of a class of integrals). Later, euler notices that due to his gamma function derivatives and integrals of fractional orders may have a meaning However, the first formal development of the subject comes in nineteenth century with the contributions of riemann, liouville, grünwald and letnikov, and since than results have been accumulated in various branches of mathematics. For Non-Integer Values of 𝒏 The Gamma function for positive no-integer values of 𝒏 the defining integral [Eq. (1)] has been evaluated numerical for 𝟏<𝒏<𝟐 and this table can be extended to all positive value of 𝒏by repeated use of Eq. (3) as shown: 𝜞𝒏−𝟏=𝜞𝒏𝒏−𝟏=𝜞𝒏+𝟏𝒏𝒏−𝟏 𝒆𝒕𝒄.