Line integral

Calculus Level 4

Find $\lfloor10000I\rfloor$ , where:

$I=\oint_Cf(z)\ \mathrm dz$

where $C$ is the square of vertices $1+i$ , $1+3i$ , $3+i$ , $3+3i$ , and $f(z)=\dfrac{e^z}{z^4-1}$ .

The answer is 0.0.

3 solutions

Lu Chee Ket
Oct 23, 2015

i --> j

Singularities z = (1 + j 0), (-1 + j 0), (0 + j) and (0 - j) are NOT enclosed by C as described, although (z^4 - 1)^n of n = 1 satisfied. (2 + j 2) is a center for rough idea.

Basically, I think those with answer j n 2 Pi i.e. j 2 Pi arise due to the fact of 2 Pi which is duplicated for zero. Otherwise, all of those coming back are just like 1 to -1 and back to 1 again in real which ought to be zero only. When angle system in degrees only valid with 0 to 360- or -180- to 180 but no greater or equals to 360, any integration coming back must only be zero. Something to do with our perspective. Nevertheless, those not enclosed are raising no doubt to contour integrals.

The answer is certainly 0.

Thushar Mn
Dec 13, 2015

by cauchy's integral theorem, if f(z) is analytic on and within a closed curve, then closed integral of f(z) on the enclosing curve=0..... <_> ..(here, f(z) depents only on z and not on conjugate of z,so f(Z) is analytic) :)

Kenny Lau
Oct 13, 2015

By Cauchy's integral theorem , since the undefined points are $i$ , $-1$ , $-i$ , $1$ , so the function is analytic everywhere, hence the integral is zero.

Line integral

The answer is 0.0.

3 solutions

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