gammafunktsiooni

Gammafunktsiooni, also known as the gamma function, is a special function in mathematics that extends the factorial function to complex numbers. It is denoted by the Greek letter gamma (Γ) and is defined for all complex numbers except the non-positive integers. The gamma function is defined by the improper integral:

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

where z is a complex number. The gamma function satisfies the functional equation:

Γ(z+1) = z * Γ(z)

This equation shows that the gamma function is a generalization of the factorial function, as for any positive integer n, Γ(n) = (n-1)!. The gamma function is widely used in various areas of mathematics, including number theory, probability, and statistics. It appears in the definition of the beta function, the Euler-Mascheroni constant, and the Riemann zeta function. The gamma function also has important applications in physics, particularly in quantum mechanics and statistical mechanics.