Chapter 4

Some Important ImproperIntegrals

4.1 Gamma FunctionThis belongs to a larger set of ideas concerning improper integrals. I will just give enoughof an introduction to this to present the very important gamma function. The Riemannintegral only is defined for bounded functions which are defined on a bounded interval.If this is not the case, then the integral has not been defined. Of course, just because thefunction is bounded does not mean the integral exists as mentioned above, but if it is notbounded, then there is no hope for it at all. However, one can consider limits of Riemannintegrals. The following definition is sufficient to deal with the gamma function in thegenerality needed in this book.

Definition 4.1.1 We say that f defined on [0,∞) is improper Riemann integrable if it isRiemann integrable on [δ ,R] for each R > 1 > δ > 0 and the following limits exist.∫

∞

0f (t)dt ≡ lim

δ→0+

∫ 1

δ

f (t)dt + limR→∞

∫ R

1f (t)dt

The gamma function is defined by

Γ(α)≡∫

∞

0e−ttα−1 dt

whenever α > 0.

Lemma 4.1.2 The limits in the above definition exist for each α > 0.

Proof: Note first that as δ → 0+, the Riemann integrals∫ 1

δ

e−ttα−1dt

increase. Thus limδ→0+∫ 1

δe−ttα−1dt either is +∞ or it will converge to the least upper

bound thanks to completeness of R. However,∫ 1

δ

tα−1dt ≤ 1α

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