About the Gamma Function

Notes for Honors Calculus II,
Originally Prepared in Spring 1995

1.  Basic Facts about the Gamma Function

The Gamma function is defined by the improper integral Γx=0txetdtt. The integral is absolutely convergent for x1 since tx1etet2,t1 and 0et2dt is convergent. The preceding inequality is valid, in fact, for all x. But for x<1 the integrand becomes infinitely large as t approaches 0 through positive values. Nonetheless, the limit limr0+r1tx1etdt exists for x>0 since tx1ettx1 for t>0, and, therefore, the limiting value of the preceding integral is no larger than that of limr0+r1tx1dt=1x. Hence, Γx is defined by the first formula above for all values x>0.

If one integrates by parts the integral Γx+1=0txetdt, writing 0udv=uvu0v00vdu, with dv=etdt and u=tx, one obtains the functional equation Γx+1=xΓx,x>0.

Obviously, Γ1=0etdt=1, and, therefore, Γ2=1·Γ1=1, Γ3=2·Γ2=2!, Γ4=3Γ3=3!, …, and, finally, Γn+1=n! for each integer n>0.

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 φ defined by φx=ex2 is the famous “bell-shaped curve” of probability theory. It can be shown that the anti-derivatives of φ are not expressible in terms of elementary functions. On the other hand, Φx=xφtdt is, by the fundamental theorem of calculus, an anti-derivative of φ, and information about its values is useful. One finds that Φ=et2dt=Γ12 by observing that et2dt=2·0et2dt, and that upon making the substitution t=u12 in the latter integral, one obtains Γ12.

To have some idea of the size of Γ12, it will be useful to consider the qualitative nature of the graph of Γx. For that one wants to know the derivative of Γ.

By definition Γx is an integral (a definite integral with respect to the dummy variable t) of a function of x and t. Intuition suggests that one ought to be able to find the derivative of Γx by taking the integral (with respect to t) of the derivative with respect to x 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 ddxtx=txlogt, one finds ddxΓx=0txlogtetdtt, and, differentiating again, d2dx2Γx=0txlogt2etdtt. One observes that in the integrals for both Γ and the second derivative Γ′′ the integrand is always positive. Consequently, one has Γx>0 and Γ′′x>0 for all x>0. This means that the derivative Γ of Γ is a strictly increasing function; one would like to know where it becomes positive.

If one differentiates the functional equation Γx+1=xΓx,x>0, one finds ψx+1=1x+ψx,x>0, where ψx=ddxlogΓx=ΓxΓx, and, consequently, ψn+1=ψ1+k=0n1k. Since the harmonic series diverges, its partial sum in the foregoing line approaches as x. Inasmuch as Γx=ψxΓx, it is clear that Γ approaches as x since Γ is steadily increasing and its integer values n1!ψn approach . Because 2=Γ3>1=Γ2, it follows that Γ cannot be negative everywhere in the interval 2x3, and, therefore, since Γ is increasing, Γ must be always positive for x3. As a result, Γ must be increasing for x3, and, since Γn+1=n!, one sees that Γx approaches as x.

It is also the case that Γx approaches as x0. To see the convergence one observes that the integral from 0 to defining Γx is greater than the integral from 0 to 1 of the same integrand. Since et1e for 0t1, one has Γx>011etx1dt=1etxxt=0t=1=1ex. It then follows from the mean value theorem combined with the fact that Γ always increases that Γx approaches as x0.

Hence, there is a unique number c>0 for which Γc=0, and Γ decreases steadily from to the minimum value Γc as x varies from 0 to c and then increases to as x varies from c to . Since Γ1=1=Γ2, the number c must lie in the interval from 1 to 2 and the minimum value Γc must be less than 1.

Image: graph of Gamma
Figure 1: Graph of the Gamma Function

Thus, the graph of Γ (see Figure 1) is concave upward and lies entirely in the first quadrant of the plane. It has the y-axis as a vertical asymptote. It falls steadily for 0<x<c to a postive minimum value Γc<1. For x>c the graph rises rapidly.

2.  Product Formulas

It will be recalled, as one may show using l'Hôpital's Rule, that et=limn1tnn. From the original formula for Γx, using an interchange of limits that in a more careful exposition would receive further comment, one has Γx=limnΓx,n, where Γx,n is defined by Γx,n=0ntx11tnndt,n1. The substitution in which t is replaced by nt leads to the formula Γx,n=nx01tx11tndt. This integral for Γx,n is amenable to integration by parts. One finds thereby: Γx,n=1xnn1x+1Γx+1,n1,n2. For the smallest value of n, n=1 , integration by parts yields: Γx,1=1xx+1. Iterating n1 times, one obtains: Γx,n=nxn!xx+1x+2x+n,n1. Thus, one arrives at the formula Γx=limnnxn!xx+1x+2x+n.

This last formula is not exactly in the form of an infinite product k=1pk=limnk=1npk. But a simple trick enables one to maneuver it into such an infinite product. One writes n as a “collapsing product”: n+1=n+1n·nn1··32·21 or n+1=k=1n1+1k, and, taking the xth power, one has n+1x=k=1n1+1kx. Since limnnxn+1x=1, one may replace the factor nx by n+1x in the last expression above for Γx to obtain Γx=1xlimnk=1n1+1kx1+xk, or Γx=1xk=11+1kx1+xk.

The convergence of this infinite product for Γx when x>0 is a consequence, through the various maneuvers performed, of the convergence of the original improper integral defining Γx for x>0.

It is now possible to represent logΓx as the sum of an infinite series by taking the logarithm of the infinite product formula. But first it must be noted that 1+tr1+rt>0fort>0,r>0. Hence, the logarithm of each term in the preceding infinite product is defined when x>0.

Taking the logarithm of the infinite product one finds: logΓx=logx+k=1ukx, where ukx=xlog1+1klog1+xk. It is, in fact, almost true that this series converges absolutely for all real values of x. The only problem with non-positive values of x lies in the fact that logx is meaningful only for x>0, and, therefore, log1+xk is meaningful only for k>|x|. For fixed x, if one excludes the finite set of terms ukx for which k|x|, 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 k=11k2. Now as k approaches , t=1k approaches 0, and so  limkukx1k2=limt0xlog1+tlog1+xtt2 =limt0x1+tx1+xt2t =limt0x1+xt1+t2t1+t1+xt =xx12. Hence, the limit of |ukxk2| is |xx12|, and the series ukx is absolutely convergent for all real x. The absolute convergence of this series foreshadows the possibility of defining Γx for all real values of x other than non-positive integers. This may be done, for example, by using the functional equation Γx+1=xΓx or Γx=1xΓx+1 to define Γx for 1<x<0 and from there to 2<x<1, etc.

Taking the derivative of the series for logΓx term-by-term – once again a step that would receive justification in a more careful treatment – and recalling the previous notation ψx for the derivative of logΓx, one obtains  ψx+1x=k=1log1+1k1k1+xk =limnk=1nlogk+1k1x+k =limnlogn+1k=1n1x+k =limnlogn+1k=1n1k+k=1n1k1x+k =limnlogn+1k=1n1k+xk=1n1kx+k =γ+xk=11kx+k, where γ denotes Euler's constant γ=limnk=1n1klogn.

When x=1 one has ψ1=1γ+k=11kk+1, and since 1kk+1=1k1k+1, this series collapses and, therefore, is easily seen to sum to 1. Hence, ψ1=γ,ψ2=ψ1+11=1γ. Since Γx=ψxΓx, one finds: Γ1=γ, and Γ2=1γ.

These course notes were prepared while consulting standard references in the subject, which included those that follow.


[1]   R. Courant, Differential and Integral Calculus (2 volumes), English translation by E. J. McShane, Interscience Publishers, New York, 1961.
[2]   E. T. Whittaker & G. N. Watson, A Course of Modern Analysis, 4th edition, Cambridge University Press, 1969.
[3]   David Widder, Advanced Calculus, 2nd edition, Prentice Hall, 1961.