Complex Analysis

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

Class 9 (Monday February 2)

SUMMARY The Complex Derivative CURRENT READING Brown & Curchill, pages 45-57 NEXT READING Brown & Curchill, pages 48-50, 55-57, 59-64

Update on Class �8 I have analyzed your comments on the Classroom Assessment Forms, and the results are: 2 mention point sets, 2 mention complex roots, 7 mention mappings, 4 mention limits and continuity (and 3 others mention no speci�c topic).

Mapping Today we will begin the class with addressing the mappings dif�culties by looking at the Joukowsky Mapping. The main thing to remember when thinking about the impact of w = f(z) on a complex set of points is that it involves a transformation of variables, from z = x + iy into w = u + iv, and that the string which ties together these two objects (i.e. the z-plane and the w-plane) is the mapping function itself f(z) = u(x, y) + iv(x, y).

Limits and Continuity You need to remember the de�nition of what a limit is (I think it is best to consult the mental image of the coordinated shrinking of neighborhoods around z_{0} and w_{0}) and how it relates to continuity. Here, also, your mathematical intuition derived from your experience with real functions should serve you well.

Example Determine the image of the circle of radius r, (r ≠ 1) under the mapping J(z) =

\frac{1}{2} (z + \frac{1}{z}).

Example Let f(z) = Arg (z), show that lim_{z→−2} Arg z does not exist.

Let f(z) = \frac{x^{2} + x}{x + y} + i\frac{y^{2} + y}{x + y}. Compute lim_{z→0}f(z).

\underline{Analyticity} If the derivative f′(z) exists at all points z of an open set G, then f is said to be analytic (or holomorphic or regular) on the set G. If f(z) is analytic on the whole complex plane, it is called entire. If �f is analytic at the point z_{0}�, what this really means is that f is analytic in a neigborhood of z_{0}. [Since �singleton� sets are closed, and not open.] When does the derivative of a function f(z) exist? What if it is written in its component form of u(x, y) and v(x, y)? Analyticity let’s us answer these questions.

Analytic functions treat the variable z as a whole unit, so that when you are given two component parts u(x, y) and v(x, y) they can always be combined to form a complex function of the single variable z = x + iy. Consider f_{1} = x^{2} − y^{2} + 2xyi and f_{2} = x^{2} − y^{2} + 3xyi By now you should be able to see that f_{1} = z^{2} while f_{2} = z^{2} + iRe(z)Im(z) f′_{1} = 2z while there is no derivative of f_{2} Cauchy-Riemann Equations Analyticity implies a relationship between the real (u(x, y)) and imaginary (v(x, y)) parts of a complex function f(z). That relationship is known as the Cauchy-Riemann Equations, which we will abbreviate C.R.E.:

u_{x} = v_{y}, u_{y} = −v_{x}

Satisfying the CRE is considered to be a necessary condition for analyticity of a function, because if f(z) is analytic then it is necessary that the CRE are also satis�ed.

ANALYTICITY ⇒ C.R.E.

To make satisfying the CRE a suf�cient condition one needs the added condition that the �rst derivatives of u and v are continuous. If both these conditions are true and f is de�ned 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}