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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84 CHAPTER 12. THE RESIDUE THEOREM AND APPLICATIONS<br />
Imz<br />
iR<br />
ir<br />
Cr<br />
CR<br />
r R<br />
Rez<br />
We note that f(z) = e ixz z α−1 is holomorphic inside the blue contour. Cauchy’s<br />
Integral Theorem thus implies that<br />
� R �<br />
0 = f(t)dt+<br />
r<br />
CR<br />
� r �<br />
f +i f(it) dt+ f.<br />
R Cr<br />
Since α < 1, we see on using Problem 12.3 (a) that<br />
��<br />
�<br />
�<br />
�<br />
CR<br />
�<br />
�<br />
f�<br />
� ≤<br />
� π/2<br />
0<br />
≤ R α−1<br />
� π/2<br />
e −xRsinθ R α−1 R dθ<br />
Likewise, since α > 0, we see that<br />
Hence<br />
� ∞<br />
0<br />
as claimed.<br />
0<br />
��<br />
�<br />
�<br />
�<br />
Cr<br />
f(t)dt = −<br />
�<br />
�<br />
f�<br />
�<br />
e −2xRθ/π α−1 π �<br />
−xR<br />
R dθ = R 1−e<br />
2x<br />
� → 0.<br />
R→∞<br />
� −xr π 1−e<br />
≤ rα<br />
2x r<br />
� 0<br />
∞<br />
f(it) idt = i α<br />
�<br />
� ∞<br />
0<br />
→<br />
r→0 0.<br />
e −xt t α−1 dt = iαΓ(α) xα ,