Question

Evaluate the fallowing integral where in case C is the circle \[\left| z \right|\] =3, counterclockwise

\[\int\limits_{C}^{}zdz/(z ^{2}+1)\]

Answer 1

@experimentX do u know cauchy teorem bro ?

Answer 2

no ... man ... i'll see

Answer 3

there are a lot of cauchy theorems which one .?

Answer 4

about contour i guess

Answer 5

can you post link ... or on wikipedia?? i'll see in a while

Answer 6

http://en.wikipedia.org/wiki/Methods_of_contour_integration @experimentX i think thats the one

Answer 7

Specifically the residue theorem: http://en.wikipedia.org/wiki/Residue_theorem

Answer 8

isn't Cauchy-Goursat theorem ?!

Answer 9

residue herp_derp is right :/ i am wrong

Answer 10

The circle encloses two simple poles (one at i, the other at -i), so their residues can be calculated by:\[R=\lim_{z \rightarrow c} (z-c) f(z)\]For c=i and c=-i. The contour integral is simply 2πi times the sum of these residues, by Cauchy's residue theorem.

Answer 11

Both residues are 1/2, so the integral is simply 2πi.

Answer 12

looks like i should go back to study and catch up with you guys!!

Answer 13

@Herp_Derp i could not understand :/

Answer 14

my english is not very well :/ maybe you can show some calculation ha ?

Answer 15

? @Herp_Derp

Answer 16

\[\int_C \frac{z~\mathrm{d}z}{(z+i)(z-i)}=2\pi i\sum_k \mathrm{Res}(f,a_k)\]\[f(z)=\frac{z}{(z+i)(z-i)},~\{a_k\}=\{-i,i\}=\{c:f~\mathrm{has~a~singularity~at~c}\}\]\[\mathrm{If~}f(z)=\frac{g(z)}{z-c}~\mathrm{for~some~g~which~is~analytic~around~}c,~\mathrm{then~Res}(f,c)=\lim_{z \rightarrow c}(z-c)f(z)=g(c).\]\[\mathrm{Therefore~Res}(f,-i)=\mathrm{Res}(f,i)=1/2~\mathrm{and}~\int_C \frac{z~\mathrm{d}z}{z^2+1}=2\pi i\]