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December 4th, 2016, 03:09 PM   #1
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Complex Integration

I am very stuck on this!

i ) i worked out the straight line is y=(x-1) does this help me, bit confused with the mod z when applying the integral.

ii) similar follows

Many thanks
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December 4th, 2016, 03:39 PM   #2
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the first parameterization is

$z = 1 + (1+i)t,~0\leq t \leq 1$

$z = (1+t)+it$

$|z|^2 = (1+t)^2 + t^2 = 2t^2 + 2t + 1$

$\displaystyle{\int_0^1}~2t^2 + 2t + 1 = \left. \dfrac{2 t^3}{3}+t^2+t\right|_0^1 =\dfrac 8 3$

for the 2nd parameterization

$z = (1+t) + t^2 i,~0\leq t \leq 1$

See if you can figure out the form of $|z|^2$ and do the integration.

Note this also goes from $1 \to (2+i)$ but by a different path.
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