Complex Analysis

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

Class 10 (Wednesday February 4)

SUMMARY Analyticity and the Cauchy-Riemann Equations CURRENT READING Brown & Curchill, pages 45-57 NEXT READING Brown & Curchill, pages 48-50, 55-57, 59-64

\underline{Groupwork}

Given g(z) = z^{2} + z + i and f(z) = \frac{1}{z} g′(z) =

f′(z) =

[g(z)f(z)]′ =

[g(z)/f(z)]′ =

Cauchy-Riemann Equations We shall derive the Cauchy-Riemann equations by looking at the definition of the derivative of a function f(z) = u(x, y) + iv(x, y) at the point z_{0}.

f′(z_{0}) = lim_{Δz→0} \frac{f(z_{0} + Δz) − f(z_{0})}{Δz}
= lim_{(Δx,Δy)→(0,0)} \frac{u(x_{0} + Δx, y_{0} + Δy) + iv(x_{0} + Δx, y_{0} + Δy) − u(x_{0}, y_{0}) − iv(x_{0}, y_{0})}{Δx + iΔy}

We shall do this limit twice, once letting Δz → 0 horizontally and the other time letting Δz → 0 vertically

This shows us that the existence of f′(z_{0}) implies the Cauchy-Riemann equations are satisfied at this point (and at every point in a neighborhood of z_{0}). This is true since if f′(z_{0}) exists then f is analytic “at” this point.

ANALYTICITY ⇒ C.R.E.

To make satisfying the CRE a sufficient condition one needs the added condition that the first derivatives of u and v are continuous. If both these conditions are true and f is defined on an open set, then f is analytic on the open set.

f′(z_{0}) = u_{x}(x_{0}, y_{0}) + iv_{x}(x_{0}, y_{0}) = −i(u_{y}(x_{0}, y_{0}) + iv_{y}(x_{0}, y_{0}))
ANALYTICITY ⟺ C.R.E. + Continuity of u_{x}, u_{y}, v_{x}, v_{y}

\underline{Example} Show that f(z) = \overline{z} is not analytic anywhere in the complex plane. You can do this in two ways: 1:

2:

\underline{Groupwork} \underline{Example} Show that the function f(z) = 1/z is analytic on the set z ≠ 0. To do that you will have to answer the following questions:

• What is its domain of definition? Is this an open set?

• What are its component functions? Are there partial derivatives continuous?

• Do they satisfy the CRE?

• Is it analytic? On what set?