This function, which for some numbers is equal to the integral from 0 to infinity of x to the t minus 1 times e to the x dx.
Numerical computations of this function often make use of the Lanczos approximation. Holder's theorem states that this function does not satisfy any algebraic differential equation with rational function coefficients. Its characterization as the unique meromorphic function which is log-convex and obeys certain functional equations is due to Bohr and Mollerup. It's not sine, but this function appears in the functional equation for the Riemann zeta function. For real arguments, it can be defined using the Euler integral of the second kind. This function has poles at all (*) nonpositive integers, but is analytic everywhere else. This function, which for some numbers is equal to the integral from 0 to infinity of x to the t minus 1 times e to the x dx. In the right half-plane, Stirling's formula provides a quick approximation to this function. For 10 points, name this function which extends the factorial function to the complex plane.
Numerical computations of this function often make use of the Lanczos approximation.
gamma function