/sci/ - Science & Math

>>9246560
You are doing something wrong because the answer should be 0 because [math]e^z[/math] is an analytic (meromorphic) function so should integrate to 0 around any closed path
you can immediately see the answer is 0, but performing the integral explicitly
you should break the integral up into three paths
[math]C = C_1 + C_2 + C_3[/math]
then
[eqn]
\oint_C e^z dz = \int_0^1 e^x dx + \int_1^{1+i} e^{1+yi} i dy + \int_{C_3} e^{x+yi} (dx + i dy) \\
= [e^x]_0^1 + [e^{1+yi}]_0^1 + \int_{1}^0 e^{(1+i)y} (1 + i) dy
[/eqn]
where in the last line I used [math]y=x[/math]
[eqn]
= (e^1 - 1) + (e^{i+1} -e^1) + [ e^{(1+i)y} ]_{1}^0 \\

= (e^1 - 1) + (e^{i+1} -e^1) + ( 1 - e^{1+i} ) \\
= 0
[/eqn]