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Home , Beta, Gamma

, , AND FUNCTIONS

There are numerous mathematical functions encountered in analysis which are defined in terms of definite integrals. Among these are the Γ(z), the beta function (n,m), and the digamma function ψ(z). We present here some of their properties including graphs and relations between them. Starting with their definitions-

∞ ∞ z −1 − t 2z −1 − u2 2 Γ(z)= ∫ t e dt = 2 ∫ u e du , where t = u 0 0

π 1 2 Β(n,m) = ∫t n (1− t)m dt = 2∫(sinθ ) 2n−1 (cosθ )2m−1dθ , where t = (sinθ )2 0 0 and

d(lnΓ(z)) 1 dΓ(z) ψ (z) = = dz Γ(z) dz

The first thing we notice, on integration by parts, is that

∞ z −t ∞ z−1 −t Γ(z +1) = −t e |0 +z∫t e dt = zΓ(z) 0

and that Γ(z+1)=z! whenever z is an integer. Taking the product of two gamma functions yields

π ∞ 2 ∞ 2 2 2 Γ(n)Γ(m) = 4∫∫dx dy[x 2m−1 2n−1e−( x + y ) ] = 4∫cosθ 2m−1 sinθ 2n−1dθ ∫ r 2(n+m)−1e −r dr 00 0 0

which upon the substitutions x=(sinθ)2 and t=r2 yields the identity

Γ(n)Γ(m) Β(n,m) = Γ(n + m)

From this last identity it follows, on setting m=n-1, that

π 2 ∞ w2n−1 π Γ(n)Γ(1 − n) = 2 (tanθ )2n+1 dθ = 2 dw = , with 0 < n <1 ∫∫2 00(1+ w ) sin(nπ )

so that Γ(1/2)= sqrt(π)=1.77245385... and (since Γ(n)=(n-1)Γ(n-1) ) that Γ(−1/2)=-2sqrt(π). Note that the integral in w above is a well known classic form encountered in complex variable theory when discussing integral evaluations involving branch points. Some of you may have noticed that in many mathematical tables the value of Γ(z) is given only for the range 1

I remember in my dissertation encountering integrals of the type

1/ 2 1/ 2 I(n) = ∫(1− 4x 2 ) n dx and J (n,m) = ∫(x 2m (1− 4x 2 ) n dx −1/ 2 −1/ 2

with n and m integer and spending considerable time evaluating them for larger values of n and m. In view of the gamma function identities above, their evaluation is actually quite simple. Using a few variable substitutions, we find

(n!) 2 4n n!Γ[m + (1/ 2)] I(n) = and J (n,m) = Γ[2(n +1)] 22m+1 Γ[n + m + (3/ 2)]

Plots of the Gamma and Beta function for a restricted range of x and y follow

For a more complete picture of the Gamma function( including its singularities at negative integers), return to the MATHFUNC page and scroll up to the area where Bessel functions are being discussed.

Next we look at the digamma function ψ(z) which we first encountered during the generation of the second solution of the Bessel equation for integer n. We start with the recurrence formula for Γ(z) given above and apply it n times to obtain

Γ(z + n) = (z)(z +1)(z + 2).....(z + n −1)Γ(z )

Next take the natural log of this expression and then take a derivative to obtain

d lnΓ(z) d lnΓ(z + n) d ln z d ln(z +1) d ln(z + n −1) ψ (z) = = − − − ..... dz dz dz dz dz

or the recurrence relation

1 1 1 ψ (z) =ψ (z + n) − − − ..... z z +1 z + n −1

which also yields the special case

1 ψ (z + 1) =ψ (z) + z

One sees at once that the function (like the gamma function) has poles at the negative integers. The following plot of ψ(z) confirms this point.

Also as z gets large the function ψ(z) goes as ln(z)-1/z , so that we can state that

m 1 ψ (1) = ln(m) − ∑ = −γ n=0 n +1 as m becomes infinite. Here γ is the well known Euler- Mascheroni constant. Using the above recurrence formula it also seen that ψ (2)=1- γ, ψ(3)=3/2− γ , and ψ(4)=11/6− γ.