We state the formula and show how it relates to the mean value property of harmonic functions, as well as prove it. We then give an example of how we can use it to compute specific difficult integrals very simply.

1. Math 3160 introduction
2. Basic Complex Algebra
3. Moduli, conjugates, triangle inequality, and polar coordinates
4. Products and quotients in exponential form
5. Roots of complex numbers
6. Functions of complex variables and mappings
7. Regions in the complex plane
8. Mappings by the exponential function
9. Limits of complex functions
10. Limits at infinity
11. The derivative of a complex function
12. Differentiation formulas for complex functions
13. Cauchy-Riemann equations
14. Analytic functions
15. Harmonic functions and analytic functions
16. The complex exponential and logarithm functions
17. Complex log identites
18. The information in analytic functions
19. Applications to signal processing
20. Applications of analytic functions to fluid flow
21. Complex exponents
22. Complex trigonometric functions
23. Inverse trigonometric functions of a complex variable
24. Derivatives and integrals of complex functions w(t)
25. Contours and arc length in the complex plane
26. Contour integrals of complex functions
27. Closed circle integral of 1/z and branch cuts
28. Moduli of complex integrals and integral bounds
29. Complex antiderivatives and the fundamental theorem
30. Proof of the antiderivative theorem for contour integrals
31. Cauchy-Goursat theorem
32. Simply and multiply connected domains
33. Cauchy integral formula
34. Cauchy Integral Results
35. The fundamental theorem of algebra revisited
36. Harmonic oscilators in the complex plane (optional)
37. How Schrodinger's equation works (optional)
38. Sequences and series involving complex variables
39. Taylor series for functions of a complex variable
40. Laurent series
41. Examples of Laurent series computations
42. Aspects of complex power series convergence
43. Singularities and residues of complex functions
44. The residue theorem
45. Residues at infinity
46. Taxonomy of singularities of complex functions
47. Aspects of zeros and poles of analytic functions
48. Zeros and poles of rational functions
49. Applications of residues to improper real integration
50. Fourier type integrals using residues