Complex Analysis

Math 312 Spring 1998  MWF 10:30am - 11:25am
Buckmire  Fowler 112

Class 21 (Friday March 6)

SUMMARY Examples of Contour Integration CURRENT READING Brown & Curchill, pages 104-122 NEXT READING Brown & Curchill pages 104-122

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:

\underline{Exercise} Consider ∫_{C} 2z^{2} dz where C is the directed line segment from z = 2 to z = −2 (Sketch the contour and evaluate the integral.)

\underline{Groupwork} Again evaluate ∫_{C} 2z^{2} dz where C is the circular arc going from z = 2 to z = −2. (Sketch the contour and evaluate the integral.)

Supplementary Exercise If you want more practice evaluating contours you should try to evaluate ∫_{C} 2z^{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.)

\underline{Question} What’s the difference between the integral ∫_{−2}^{2} 2\overline{z}^{2} dz and ∫_{−2}^{2} 2z^{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?

Proprties of Contour Integrals

∫_{C} f(z)dz = ∫_{C_{1}} f(z)dz + ∫_{C_{2}} f(z)dz
∫_{−C} f(z)dz = − ∫_{C} f(z)dz
∫_{C} z_{0}f(z)dz = z_{0} ∫_{C} f(z)dz, z_{0} ∈ C
∫_{C} f(z) + g(z)dz = ∫_{C} f(z)dz + ∫_{C} g(z)dz

|∫_{C} f(z)dz| ≤ ML

where L is the length of the contour and ∫_{a}^{b} |z′(t)|dt ≤ L and M is an upper bound on f(z), |f(z)| ≤ M \underline{Exercise} If is the arc of the circle |z| = 2 traversed in the counter-clockwise direction, then we want to show that

|∫ \frac{e^{z}}{z^{2} + 1}dz| ≤ \frac{4πe^{2}}{3}