About the Gamma Function

Transcription

1 About the Gamma Function Notes for Honors Calculus II, Originally Prepared in Spring 995 Basic Facts about the Gamma Function The Gamma function is defined by the improper integral Γ) = The integral is absolutely convergent for since t t dt e t t e t e t/2, t and e t/2 dt is convergent The preceding inequality is valid, in fact, for all But for < the integrand becomes infinitely large as t approaches through positive values Nonetheless, the limit lim r + t e t dt eists for > since r t e t t for t >, and, therefore, the limiting value of the preceding integral is no larger than that of lim r + t dt = Hence, Γ) is defined by the first formula above for all values > If one integrates by parts the integral writing r Γ + ) = t e t dt, udv = u )v ) u)v) with dv = e t dt and u = t, one obtains the functional equation Γ + ) = Γ), > vdu, Obviously, Γ) = e t dt =, and, therefore, Γ2) = Γ) =, Γ3) = 2 Γ2) = 2!, Γ4) = 3Γ3) = 3!,, and, finally, Γn + ) = n!

2 for each integer n > 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 ϕ) = e 2 is the famous bell-shaped curve of probability theory It can be shown that the anti-derivatives of ϕ are not epressible in terms of elementary functions On the other hand, Φ) = ϕt)dt is, by the fundamental theorem of calculus, an anti-derivative of ϕ, and information about its values is useful One finds that by observing that Φ ) = e t2 dt = 2 e t2 dt = Γ/2) e t2 dt, and that upon maing the substitution t = u /2 in the latter integral, one obtains Γ/2) To have some idea of the size of Γ/2), it will be useful to consider the qualitative nature of the graph of Γ) For that one wants to now the derivative of Γ By definition Γ) is an integral a definite integral with respect to the dummy variable t) of a function of and t Intuition suggests that one ought to be able to find the derivative of Γ) by taing the integral with respect to t) of the derivative with respect to of the integrand Unfortunately, there are eamples 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, d d Γ) = d d t = t log t), t t dt log t)e t, d 2 d 2 Γ) = t log t) 2 t dt e t One observes that in the integrals for both Γ and the second derivative Γ the integrand is always positive Consequently, one has Γ) > and Γ ) > for all > This means that the derivative Γ of Γ is a strictly increasing function; one would lie to now where it becomes positive 2

3 If one differentiates the functional equation Γ + ) = Γ), >, one finds ψ + ) = + ψ), >, where and, consequently, ψ) = d d log Γ) = Γ ) Γ), ψn + ) = ψ) + Since the harmonic series diverges, its partial sum in the foregoing line approaches as Inasmuch as Γ ) = ψ)γ), it is clear that Γ approaches as since Γ is steadily increasing and its integer values n )!ψn) approach Because 2 = Γ3) > = Γ2), it follows that Γ cannot be negative everywhere in the interval 2 3, and, therefore, since Γ is increasing, Γ must be always positive for 3 As a result, Γ must be increasing for 3, and, since Γn + ) = n!, one sees that Γ) approaches as It is also the case that Γ) approaches as To see the convergence one observes that the integral from to defining Γ) is greater than the integral from to of the same integrand Since e t /e for t, one has Γ) > n = [ t /e)t dt = /e) ] t = t= = e It then follows from the mean value theorem combined with the fact that Γ always increases that Γ ) approaches as Hence, there is a unique number c > for which Γ c) =, and Γ decreases steadily from to the minimum value Γc) as varies from to c and then increases to as varies from c to Since Γ) = = Γ2), the number c must lie in the interval from to 2 and the minimum value Γc) must be less than Y X Figure : Graph of the Gamma Function Thus, the graph of Γ see Figure ) is concave upward and lies entirely in the first quadrant of the plane It has the y-ais as a vertical asymptote It falls steadily for < < c to a postive minimum value Γc) < For > c the graph rises rapidly 3

4 2 Product Formulas It will be recalled, as one may show using l Hôpital s Rule, that e t = lim t n n) From the original formula for Γ), using an interchange of limits that in a more careful eposition would receive further comment, one has Γ) = lim Γ, n), where Γ, n) is defined by Γ, n) = n t t n) n dt, n The substitution in which t is replaced by nt leads to the formula Γ, n) = n t t) n dt This integral for Γ, n) is amenable to integration by parts One finds thereby: Γ, n) = ) + n Γ +, n ), n 2 n For the smallest value of n, n =, integration by parts yields: Γ, ) = + ) Iterating n times, one obtains: Thus, one arrives at the formula Γ, n) = n n! + ) + 2) + n), n Γ) = n! lim n + ) + 2) + n) This last formula is not eactly in the form of an infinite product p = lim n p But a simple tric enables one to maneuver it into such an infinite product One writes n as a collapsing product : n + = n + n n n

5 or n + = and, taing the th power, one has n + ) = n n + ), + ) Since lim n n + ) =, one may replace the factor n by n + ) in the last epression above for Γ) to obtain or Γ) = n + lim ) +, Γ) = ) + ) + ) The convergence of this infinite product for Γ) when > is a consequence, through the various maneuvers performed, of the convergence of the original improper integral defining Γ) for > It is now possible to represent log Γ) as the sum of an infinite series by taing the logarithm of the infinite product formula But first it must be noted that + t) r + rt > for t >, r > Hence, the logarithm of each term in the preceding infinite product is defined when > Taing the logarithm of the infinite product one finds: log Γ) = log + u ), where u ) = log + ) log + ) 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 log) is meaningful only for >, and, therefore, log + /) is meaningful only for > For fied, if one ecludes the finite set of terms u ) 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 eists and is finite For this one may tae as the test series, the series 2 5

6 Now as approaches, t = / approaches, and so u ) log + t) log + t) lim / 2 = lim t t 2 +t = lim +t t 2t [ + t) + t)] = lim t 2t + t) + t) ) = 2 Hence, the limit of u )/ 2 is )/2, and the series u ) is absolutely convergent for all real The absolute convergence of this series foreshadows the possibility of defining Γ) for all real values of other than non-positive integers This may be done, for eample, by using the functional equation Γ + ) = Γ) or Γ) = Γ + ) to define Γ) for < < and from there to 2 < <, etc Taing the derivative of the series for log Γ) term-by-term once again a step that would receive justification in a more careful treatment and recalling the previous notation ψ) for the derivative of log Γ), one obtains ψ) + = = lim log n + ) ) + log + + = lim logn + ) = lim logn + ) = lim = γ + where γ denotes Euler s constant γ = logn + ) lim + ), n n + n n + ) + n n + + ) ) log n When = one has ψ) = γ + + ), 6

7 and since + ) = +, this series collapses and, therefore, is easily seen to sum to Hence, Since Γ ) = ψ)γ), one finds: ψ) = γ, ψ2) = ψ) + / = γ Γ ) = γ, and Γ 2) = γ These course notes were prepared while consulting standard references in the subject, which included those that follow References [] R Courant, Differential and Integral Calculus 2 volumes), English translation by E J McShane, Interscience Publishers, New Yor, 96 [2] E T Whittaer & G N Watson, A Course of Modern Analysis, 4th edition, Cambridge University Press, 969 [3] David Widder, Advanced Calculus, 2nd edition, Prentice Hall, 96 7