The Gamma function is defined by the improper integral The integral is absolutely convergent for since and is convergent. The preceding inequality is valid, in fact, for all . But for the integrand becomes infinitely large as approaches through positive values. Nonetheless, the limit exists for since for , and, therefore, the limiting value of the preceding integral is no larger than that of Hence, is defined by the first formula above for all values .
If one integrates by parts the integral writing with and , one obtains the functional equation
Obviously, , and, therefore, , , , …, and, finally, for each integer .
Thus, the gamma function provides a way of giving a meaning to the “factorial” of any positive real number.
Another reason for interest in the gamma function is its relation to integrals that arise in the study of probability. The graph of the function defined by is the famous “bell-shaped curve” of probability theory. It can be shown that the anti-derivatives of are not expressible in terms of elementary functions. On the other hand, is, by the fundamental theorem of calculus, an anti-derivative of , and information about its values is useful. One finds that by observing that and that upon making the substitution in the latter integral, one obtains .
To have some idea of the size of , it will be useful to consider the qualitative nature of the graph of . For that one wants to know the derivative of .
By definition is an integral (a definite integral with respect to the dummy variable ) of a function of and . Intuition suggests that one ought to be able to find the derivative of by taking the integral (with respect to ) of the derivative with respect to of the integrand. Unfortunately, there are examples where this fails to be correct; on the other hand, it is correct in most situations where one is inclined to do it. The methods required to justify “differentiation under the integral sign” will be regarded as slightly beyond the scope of this course. A similar stance will be adopted also for differentiation of the sum of a convergent infinite series.
Since one finds and, differentiating again, One observes that in the integrals for both and the second derivative the integrand is always positive. Consequently, one has and for all . This means that the derivative of is a strictly increasing function; one would like to know where it becomes positive.
If one differentiates the functional equation one finds where and, consequently, Since the harmonic series diverges, its partial sum in the foregoing line approaches as . Inasmuch as , it is clear that approaches as since is steadily increasing and its integer values approach . Because , it follows that cannot be negative everywhere in the interval , and, therefore, since is increasing, must be always positive for . As a result, must be increasing for , and, since , one sees that approaches as .
It is also the case that approaches as . To see the convergence one observes that the integral from to defining is greater than the integral from to of the same integrand. Since for , one has It then follows from the mean value theorem combined with the fact that always increases that approaches as .
Hence, there is a unique number for which , and decreases steadily from to the minimum value as varies from to and then increases to as varies from to . Since , the number must lie in the interval from to and the minimum value must be less than .
Figure 1: Graph of the Gamma Function
Thus, the graph of (see Figure 1) is concave upward and lies entirely in the first quadrant of the plane. It has the -axis as a vertical asymptote. It falls steadily for to a postive minimum value . For the graph rises rapidly.
It will be recalled, as one may show using l'Hôpital's Rule, that From the original formula for , using an interchange of limits that in a more careful exposition would receive further comment, one has where is defined by The substitution in which is replaced by leads to the formula This integral for is amenable to integration by parts. One finds thereby: For the smallest value of , , integration by parts yields: Iterating times, one obtains: Thus, one arrives at the formula
This last formula is not exactly in the form of an infinite product But a simple trick enables one to maneuver it into such an infinite product. One writes as a “collapsing product”: or and, taking the th power, one has Since one may replace the factor by in the last expression above for to obtain or
The convergence of this infinite product for when is a consequence, through the various maneuvers performed, of the convergence of the original improper integral defining for .
It is now possible to represent as the sum of an infinite series by taking the logarithm of the infinite product formula. But first it must be noted that Hence, the logarithm of each term in the preceding infinite product is defined when .
Taking the logarithm of the infinite product one finds: where It is, in fact, almost true that this series converges absolutely for all real values of . The only problem with non-positive values of lies in the fact that is meaningful only for , and, therefore, is meaningful only for . For fixed , if one excludes the finite set of terms for which , then the remaining “tail” of the series is meaningful and is absolutely convergent. To see this one applies the “ratio comparison test” which says that an infinite series converges absolutely if the ratio of the absolute value of its general term to the general term of a convergent positive series exists and is finite. For this one may take as the “test series”, the series Now as approaches , approaches , and so Hence, the limit of is , and the series is absolutely convergent for all real . The absolute convergence of this series foreshadows the possibility of defining for all real values of other than non-positive integers. This may be done, for example, by using the functional equation or to define for and from there to , etc.
Taking the derivative of the series for term-by-term – once again a step that would receive justification in a more careful treatment – and recalling the previous notation for the derivative of , one obtains where denotes Euler's constant
When one has and since this series collapses and, therefore, is easily seen to sum to . Hence, Since , one finds: and
These course notes were prepared while consulting standard references in the subject, which included those that follow.