Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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12.2. THE GAMMA FUNCTION 83<br />
Since α < 1, we see that the contribution from the circular arc CR is<br />
��<br />
�<br />
� �<br />
�<br />
� f�<br />
�<br />
CR<br />
≤ Rα−1<br />
R<br />
2<br />
·2πR = 4πR α−1 →<br />
R→∞ 0.<br />
Likewise, since α > 0, the contribution from the semicircular contour Cr is<br />
��<br />
�<br />
� �<br />
�<br />
� f�<br />
�<br />
We thus deduce that<br />
Thus<br />
Cr<br />
2πie (α−1)iπ = lim<br />
r→0<br />
≤ rα−1<br />
1<br />
2<br />
R→∞ ir<br />
� ∞<br />
e (α−1)log|z|<br />
·πr = 2πr α →<br />
r→0 0.<br />
�� R+ir � R−ir<br />
f −<br />
=<br />
0 +<br />
dz −<br />
1+z<br />
= I(α) � 1−e (α−1)2πi� .<br />
�<br />
f<br />
−ir<br />
� ∞<br />
e (α−1)(log|z|+i2π)<br />
0 +<br />
1+z<br />
π = I(α)· e−(α−1)πi −e (α−1)πi<br />
= I(α)·<br />
2i<br />
−e−απi +eαπi ,<br />
2i<br />
from which we see that<br />
I(α) = Γ(α)Γ(1−α) = π<br />
sinπα ,<br />
On extending this result by analytic continuation, one finds for all z ∈ C \ Z<br />
that<br />
Γ(z)Γ(1−z) = π<br />
sinπz .<br />
• For α ≥ 1 and positive x and λ, another frequently encountered integral can be<br />
expressed in terms <strong>of</strong> Γ using the substitution u = xtλ :<br />
�<br />
� ∞<br />
0<br />
e −xtλ<br />
t α−1 dt = 1<br />
λx α<br />
λ<br />
0<br />
� ∞<br />
0<br />
e −u u α<br />
λ−1 du = Γ� α<br />
λ<br />
λx α<br />
λ<br />
For the special case α = x = 1, this result simplifies to<br />
� ∞<br />
e −tλ<br />
dt = 1<br />
λ Γ<br />
� � �<br />
1<br />
= Γ 1+<br />
λ<br />
1<br />
�<br />
.<br />
λ<br />
For 0 < α < 1 and x > 0, a related integral is<br />
� ∞<br />
0<br />
e ixt t α−1 dt = iα Γ(α)<br />
x α .<br />
dz<br />
. (12.3)<br />
(12.4)<br />
To establish this result, it is convenient to introduce a branch cut, shown in<br />
red, along the negative real axis: