RESIDUE CALCULUS, PART II

[Pages:14]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 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download