Incomplete Gamma Function - Indefinite and Definite Integrals

Indefinite and Definite Integrals

The following indefinite integrals are readily obtained using integration by parts:

\int x^{b-1} \gamma(s,x) \mathrm d x = \frac{1}{b} \left( x^b \gamma(s,x) + \Gamma(s+b,x) \right).
\int x^{b-1} \Gamma(s,x) \mathrm d x = \frac{1}{b} \left( x^b \Gamma(s,x) - \Gamma(s+b,x) \right),

The lower and the upper incomplete Gamma function are connected via the Fourier transform:

\int_{-\infty}^\infty \frac {\gamma\left(\frac s 2, z^2 \pi \right)} {(z^2 \pi)^\frac s 2} e^{-2 \pi i k z} \mathrm d z = \frac {\Gamma\left(\frac {1-s} 2, k^2 \pi \right)} {(k^2 \pi)^\frac {1-s} 2}.

This follows, for example, by suitable specialization of (Gradshteyn & Ryzhik 1980, § 7.642).

Read more about this topic:  Incomplete Gamma Function

Famous quotes containing the words indefinite and/or definite:

    “As they are not seen on their way down the streams, it is thought by fishermen that they never return, but waste away and die, clinging to rocks and stumps of trees for an indefinite period; a tragic feature in the scenery of the river bottoms worthy to be remembered with Shakespeare’s description of the sea-floor.”
    —Henry David Thoreau (1817–1862)

    “Our tradition of political thought had its definite beginning in the teachings of Plato and Aristotle. I believe it came to a no less definite end in the theories of Karl Marx.”
    —Hannah Arendt (1906–1975)