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Although the main definition of the gamma function—the Euler integral of the second kind—is only valid (on the real axis) for positive arguments, its domain can be extended with analytic continuation [13] to negative arguments by shifting the negative argument to positive values by using either the Euler's reflection formula ...
In mathematics, a reflection formula or reflection relation for a function f is a relationship between f(a − x) and f(x). It is a special case of a functional equation . It is common in mathematical literature to use the term "functional equation" for what are specifically reflection formulae.
The classical gamma function satisfies the functional equation (+) = for any .This has an analogue with respect to the Morita gamma function: (+) = {,,.The Euler's reflection formula () = has its following simple counterpart in the p-adic case:
The roots of the digamma function are the saddle points of the complex-valued gamma function. Thus they lie all on the real axis. The only one on the positive real axis is the unique minimum of the real-valued gamma function on R + at x 0 = 1.461 632 144 968 362 341 26.... All others occur single between the poles on the negative axis:
The color of a point encodes the value of the function. Darker colors denote values closer to zero and hue encodes the value's argument. In mathematics, the Riemann xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation.
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.
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
the gamma function, a generalization of the factorial [2] the upper incomplete gamma function; the modular group, the group of fractional linear transformations; the gamma distribution, a continuous probability distribution defined using the gamma function; second-order sensitivity to price in mathematical finance