Gammaderivations

Gammaderivations are a class of mathematical functions that are derived from the gamma function, a special function in mathematics that extends the factorial function to complex numbers. The gamma function, denoted by Γ(z), is defined for all complex numbers z except for the non-positive integers, where it has simple poles. It is defined by the improper integral:

Γ(z) = ∫ from 0 to ∞ of t^(z-1) * e^(-t) dt

Gammaderivations refer to functions that are obtained by differentiating the gamma function with respect to its argument. The first derivative of the gamma function is denoted by Γ'(z) and is known as the digamma function, denoted by ψ(z) or ψ₁(z). The digamma function is important in various areas of mathematics and physics, including number theory, statistics, and quantum mechanics.

Higher-order derivatives of the gamma function are also known as polygamma functions. The nth derivative of the gamma function is denoted by Γ^(n)(z) and is known as the nth polygamma function, denoted by ψ(n)(z) or ψ_n(z). These functions are defined for all complex numbers z except for the non-positive integers, where they have poles.

Gammaderivations have many interesting properties and applications. For example, they are related to the Riemann zeta function and the Hurwitz zeta function, which are important in number theory. They also appear in the study of special functions, such as the polylogarithm function and the Lerch transcendent.

In summary, gammaderivations are a class of mathematical functions that are derived from the gamma function. They include the digamma function and the polygamma functions, which have many interesting properties and applications in various areas of mathematics and physics.