Complex Analysis

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

Class 8 (Friday January 30)

SUMMARY Continuity and Differentiability of Complex Functions CURRENT READING Brown & Curchill, pages 40-45 NEXT READING Brown & Curchill, pages 45-57

Continuity

A complex function f(z) is continuous at a point z_{0} if all three of the following statements are true

1: lim_{z→z0}f(z) exists

2: f(z_{0}) exists

3: lim_{z→z0}f(z) = f(z_{0})

Consider the function below:

f(z) = \{ \begin{matrix}\frac{z^{2} + 4}{z − 2i}, & z ≠ 2i \\ 3 + 4i, & z = 2i \end{matrix}

Answer the following questions

(a) What is the value of lim_{z→2i}f(z)?

(b) Is f(z) continuous at z = 2i?

(c) Is f(z) continuous at points z ≠ 2i?

We say that the function f(z) defined above has a removable singularity at z = 2i. Write down the definition of f(z) which has had the singularity removed.

More Aspects of Continuity As with real functions of a real variable, sums, differences, products and compositions of continuous functions are continuous.

When f(z) continuous ⟺ u(x, y) and v(x, y) continuous

When f(z) continouus in a region R, then |f(z)| is also continuous in the region R and if R is a bounded and closed set then there exists a positive number M so that |f(z)| ≤ M∀z ∋ R.

Derivative

Let f be defined in a neigborhood around z_{0}. The derivative of f at z_{0}, denoted by f′(z_{0}), is defined by

f′(z_{0}) = lim_{z→z0} \frac{f(z) − f(z_{0})}{z − z_{0}}

provided the above limit exists. The function f is said to be differentiable at z_{0}. Consider f(z) = z^{2}. Write down the expression \frac{Δw}{Δz} = \frac{f(z+Δz)−f(z)}{Δz}

The derivative \frac{dw}{dz} = f′(z) is defined as f′(z) = lim_{Δz→0} \frac{Δw}{Δz} Evaluate this limit for our function f(z) = z^{2}.

Write down f′(z)

Write down the real and imaginary parts of the function f(z) = z^{2}

Write down the real and imaginary parts of the function f′(z) See any patterns?

Rules of Differentiation

The standard rules of differentiating function that you learned for real functions basically apply to complex functions. Scilicet:

\frac{d}{dz} (c) = 0  \frac{d}{dz} (z) = 1  \frac{d}{dz} (z^{n}) = nz^{n−1}  \frac{d}{dz} (e^{z}) = e^{z}

Linearity

\frac{d}{dz} [cf(z) + g(z)] = cf′(z) + g′(z) c constant

Product Rule

\frac{d}{dz} [f(z)g(z)] = f′(z)g(z) + f(z)g′(z)

Quotient Rule

\frac{d}{dz} [\frac{f(z)}{g(z)}] = \frac{f′(z)g(z) − f(z)g′(z)}{(g(z))^{2}}

Aspects of Differentiation One of the most important aspects to remember about differentiability and continuity is:

DIFFERENTIABILITY ⇒ CONTINUITY

CONTINUITY DOES NOT IMPLY DIFFERENTIABILITY.