1. ECE 6382 Notes 2 Differentiation of Functions of a Complex Variable
Fall 2017
David R. Jackson
Notes 2
Differentiation of Functions of a Complex Variable
Notes are adapted from D. R. Wilton, Dept. of ECE

2. Functions of a Complex Variable

3. Differentiation of Functions of a Complex Variable

4. The Cauchy – Riemann Conditions

5. The Cauchy – Riemann Conditions (cont.)

6. The Cauchy – Riemann Conditions (cont.)

7. The Cauchy – Riemann Conditions (cont.)
Hence we have the following equivalent statements:

8. The Cauchy – Riemann Conditions (cont.)

9. Applying the Cauchy – Riemann Conditions

10. Applying the Cauchy – Riemann Conditions (cont.)

11. Applying the Cauchy – Riemann Conditions (cont.)

12. Differentiation Rules

13. Differentiation Rules (cont.)

14. Differentiation Rules

15. A Theorem Related to z* If f = f (z,z*) is analytic, then
(The function cannot really vary with z*.)

16. A Theorem Related to z* (cont.)

17. A Theorem Related to z* (cont.)

18. Analytic Functions
A function that is analytic everywhere is called “entire”.
Composite functions of analytic functions are also analytic.
Derivatives of analytic functions are also analytic. (This is proven later.)

19. Real and Imaginary Parts of Analytic Functions Are Harmonic Functions
This result is extensively used
in conformal mapping to solve
electrostatics and other problems
involving the 2D Laplace equation
(discussed later).

20. Real and Imaginary Parts of Analytic Functions Are Harmonic Functions
Proof
Assume df/dz is also analytic (see note on previous slide)!