MAT 3730 Complex Variables Section 2.4 Cauchy Riemann Equations

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

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 z 0 is that the C-R eq. hold at z 0. Consequently, if f is analytic in an open set G, then the C-R eq. must hold at every point of G.

A necessary condition for a fun. f(z)=u(x,y)+iv(x,y) to be diff. at a point z 0 is that the C-R eq. hold at z 0. Consequently, if f is analytic in an open set G, then the C-R eq. must hold at every point of G. Theorem 4

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

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

Let f(z)=u(x,y)+iv(x,y) be defined in some open set G containing the point z 0. If the first partial derivatives of u and v 1. exist in G 2. are continuous at z 0, and 3. satisfy the C-R eq. at z 0 Then f is differentiable at z 0. Theorem 5 Part I (Sufficient Conditions)

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. Theorem 5 Part II

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

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

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.

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