Multiplication Theorem - Gamma Function-Legendre Function

Gamma Function-Legendre Function

The duplication formula and the multiplication theorem for the gamma function are the prototypical examples. The duplication formula for the gamma function is

\Gamma(z) \; \Gamma\left(z + \frac{1}{2}\right) = 2^{1-2z} \; \sqrt{\pi} \; \Gamma(2z). \,\!

It is also called the Legendre duplication formula or Legendre relation, in honor of Adrien-Marie Legendre. The multiplication theorem is

\Gamma(z) \; \Gamma\left(z + \frac{1}{k}\right) \; \Gamma\left(z + \frac{2}{k}\right) \cdots \Gamma\left(z + \frac{k-1}{k}\right) = (2 \pi)^{ \frac{k-1}{2}} \; k^{1/2 - kz} \; \Gamma(kz) \,\!

for integer k ≥ 1, and is sometimes called Gauss's multiplication formula, in honour of Carl Friedrich Gauss. The multiplication theorem for the gamma functions can be understood to be a special case, for the trivial character, of the Chowla–Selberg formula.

Read more about this topic:  Multiplication Theorem

Famous quotes containing the word function:

    “Philosophical questions are not by their nature insoluble. They are, indeed, radically different from scientific questions, because they concern the implications and other interrelations of ideas, not the order of physical events; their answers are interpretations instead of factual reports, and their function is to increase not our knowledge of nature, but our understanding of what we know.”
    —Susanne K. Langer (1895–1985)