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\centerline{{\it  Class 21} {\bf (Friday March 6)}}
 
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\item[SUMMARY] Examples of Contour Integration
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\item[CURRENT READING] Brown \& Curchill, pages 104-122
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\item[NEXT READING] Brown \& Curchill pages 104-122
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{\bf Parametricization of a line segment}\\
Can you come up with a formula for a parametrization $z(t)$ of a directed line segment from point $z_1$ to $z_2$ where $t$ starts at $t_1$ and ends at $t_2$ so that $z(t_1) = z_1$ and $z(t_2) = z_2$? Write it down below:

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\underline{\bf Exercise}\\
Consider ${\displaystyle \int_C 2 z^2 \, dz}$ where $C$ is the directed line segment from $z=2$ to $z=-2$ (Sketch the contour and evaluate the integral.)
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\underline{\sc Groupwork}\\
Again evaluate ${\displaystyle \int_C 2 z^2 \, dz}$ where $C$ is the  circular arc going from $z=2$ to $z=-2$. (Sketch the contour and evaluate the integral.)
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\underline{\sc Supplementary Exercise}\\
If you want more practice evaluating contours you should try to
evaluate ${\displaystyle \int_C 2 z^2 \, dz}$ where $C$ is the  directed line segment going from $z=2$ to $z=-2$ via the point $z=2i$. (Sketch the contour and evaluate the integral.) 
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\underline{\bf Question}\\
What's the difference between the integral ${\int_{-2}^2 2 \zc^2 \ dz}$ and $\int_{-2}^2 2 z^2 \ dz$?\\ 
Does the value of a contour integral depend on the path taken?\\

Does path dependence of a contour integral depend on the function involved? What property of the function is involved?\\[14pt]

{\bf Proprties of Contour Integrals}

$${\displaystyle \int_C f(z) dz =  \int_{C_1} f(z) dz + \int_{C_2} f(z) dz }$$

$${\displaystyle \int_{-C} f(z) dz = - \int_C f(z) dz } $$

$${\displaystyle \int_{C} z_0 f(z) dz = z_0 \int_C f(z) dz, } \qquad z_0 \in {\bf C}$$

$${\displaystyle \int_{C}  f(z) + g(z) dz =  \int_C f(z) dz + \int_C g(z) dz}$$

$${\displaystyle \left| \int_{C}  f(z) dz  \right| } \le ML$$

where $L$ is the length of the contour and ${\displaystyle  \int_a^b |z'(t)| dt} \le L$\\
 and $M$ is an upper bound on $f(z)$, $| f(z) | \le M$

\underline{\bf Exercise}\\
If $\Gamma$ is the arc of the circle $|z|=2$ traversed in the counter-clockwise
direction, 
then we want to show that
$${\displaystyle \left| \int_{\Gamma} \frac{e^z}{z^2+1} dz  \right|} \le 
{\displaystyle \frac{4 \pi e^2}{3}}$$

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