Partial Differentials Notes_Part_7

Partial Differentials Notes_Part_7 - -4-551015Figure...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: -4-224-15-10-551015Figure 12.3.The Gamma Function.Thus, at integer values ofx, the gamma function agrees with the elementary factorial.A few other values can be computed exactly. One important case is whenx=12. Usingthe substitutiont=s2, withdt= 2s ds, we findΓ(12)=integraldisplay∞e−tt−1/2dt=integraldisplay∞2e−s2dx=√π,(12.63)where the final integral was evaluated in (Gaussint). Thus, using the identification withthe factorial function, we identify this value with(−12)! =√π. The recurrence relation(12.61) will then fix the value of the gamma function at all half-integers12,32,52, . . .. Forexample,Γ(32)=12Γ(12)=12√π,(12.64)and hence12! =12√π.Moreover, one can use the recurrence formula (12.61) to extend the definition of Γ(x)to all non-integralx <−1. For example,Γ(12)=12Γ(−12),soΓ(−12)= Γ(−32)=−2√π ,whileΓ(−12)=−12Γ(−32),soΓ(−12)= Γ(−32)=43√π ,and so on. The only points at which this device fails is whenxis a negative integer,and indeed, Γ(x) has a singularity whenx=−1,−2,−3, . . .. A graph of the gammafunction appear in Figure 12.3. With some further work, Γ(z) can be extended to ananalytic function defined on the entire complex plane,z∈C, except for simple poles atthe negative integers,z=−1,−2,−3, . . .....
View Full Document

This note was uploaded on 02/10/2012 for the course MATH 5587 taught by Professor Olver during the Fall '10 term at University of Central Florida.

Page1 / 3

Partial Differentials Notes_Part_7 - -4-551015Figure...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online