MAT 3730 Complex Variables

MAT 3730Complex Variables Section 2.4 Cauchy Riemann Equations http://myhome.spu.edu/lauw

Preview • Necessary and Sufficient conditions for a function to be differentiable at a point. • Introduce the Cauchy-Riemann Equations For real function f, • Look at the corresponding result in complex functions

Cauchy-Riemann Equations

Cauchy-Riemann Equations • We have proved the following theorem.

Theorem 4 A necessary condition for a fun. f(z)=u(x,y)+iv(x,y) to be diff. at a point z0 is that the C-R eq. hold at z0. Consequently, if f is analytic in an open set G, then the C-R eq. must hold at every point of G.

Application of Theorem 4 To show that a function is NOT analytic, it suffices to show that the C-R eq. are not satisfied

Example 1 Show that the function is not analytic at any point.

Theorem 5 Part I (Sufficient Conditions) Let f(z)=u(x,y)+iv(x,y) be defined in some open set G containing the point z0. If the first partial derivatives of u and v 1. exist in G 2. are continuous at z0, and 3. satisfy the C-R eq. at z0 Then f is differentiable at z0.

Theorem 5 Part II Consequently, if the first partial derivatives are continuous and satisfy the C-R eq. at all points of G, then f is analytic in G.

Guess? Example 2 Prove that the function is entire, and find its derivative

Example 2 Prove that the function is entire, and find its derivative Suffices to show f is diff. everywhere Suffices to show (a) all partial derivatives exist and cont. everywhere (b) f satisfy the C-R eq. everywhere

Theorem 6 If (a) f(z) is analytic in a domain D (b) f’(z)=0 everywhere in D then f(z) is constant in D.

Recall: Theorem (Section 1.6)

Theorem 6 If (a) f(z) is analytic in a domain D (b) f’(z)=0 everywhere in D then f(z) is constant in D.

Next Class • Fast Read Section 2.5

MAT 3730 Complex Variables