Contour Integration
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Contour Integration
Calculate the integral of the functions in the following paths:
a) \( f(z)=z \), along the path from the origin to point 1+i.
b) \( f(z)=z \), along the path formed by two segments, the first segment going from the
origin to the point (1, 0), and the second segment going from the point (1, 0) to the point (1, 1).
c) \( f(z)=\bar{z} \), along the path from the origin to point 1+i.
d) \( f(z)=\bar{z} \), along the path formed by two segments, the first segment going from
the origin to the point (1, 0), and the second segment going from the point (1, 0) to the point (1, 1).
e) \( f(z)=y-x^{2} \), along the path formed by two segments, the first segment going from
the origin to the point (2, 0), and the second segment going from the point (2, 0) to the point (2, 1).
f) \( f(z)=y-x^{2} \), along the path formed by two segments, the first segment going
from the origin to the point (0, 1), and the second segment going from the point (0, 1) to the point (2, 1).
g) \( f(z)=x^{2}-y^{2}+i(x-y^{2}) \), along the path from the origin to point 3+2i.
h) \( z=x^{2}y+3xyi \), along the path from the point 1+i to point 2+3i.
Calculate the integral of the function
\( \displaystyle f(z)=\frac{z+2}{z} \)
along the following paths
a) along the semicircle
\( z=2\operatorname{e}^{\mathsf{i}\theta} \),
\( (0\leqslant \theta\leqslant \pi) \).
b) along the semicircle
\( z=2\operatorname{e}^{\mathsf{i}\theta} \),
\( (\pi \leqslant \theta\leqslant 2\pi) \).
c) along the circle
\( z=2\operatorname{e}^{\mathsf{i}\theta} \),
\( (0\leqslant \theta\leqslant 2\pi) \).
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