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**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, . . .....

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