RESIDUE CALCULUS, PART II
Lecture 9
RESIDUE CALCULUS, PART II
Applications: Contour integrals in the presence of branch cuts
Summation of series by residue calculus
CONTOUR INTEGRALS IN THE PRESENCE OF BRANCH CUTS
? require combining techniques for isolated singular points, e.g. residue theorem,
with techniques for branch points
Integral of the square root round the unit circle
Take principal branch :
f (z)
=
z
=
rei/2
,
0 < 2 . Branch cut along R+ .
? can't apply Cauchy theorem to |z| = 1 but can apply it to contour :
Im z
f (z) dz = 0
- 1
? Then write = + + +
C1
L1
C
L2
C1
C
L2
1
L1
Re z
Let 0 . By Darboux inequality |
z dz| 2
-0
0.
C
Thus
z dz = -
0
dx
x
ei2/2
-
1 dx x = -2
1 dx x = -4/3 .
C1
1
0
0
CALCULATION OF
I=
dx
0
xp-1 x2 + 1
,
0 ................
................
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