Complex Analysis

  1. What is the Comparison Test?
    Let Image Upload 1 be a convergent series of real number. Image Upload 2 for all Image Upload 3. If |Image Upload 4 then Image Upload 5 converges absolutely.
  2. What is Morera's Theorem? 
    (Converse of Cauchy- Goursat)
    Let Image Upload 6 be continuous on a domain Image Upload 7 if Image Upload 8 for all simple closed contours in Image Upload 9 then Image Upload 10 is analytic on D
  3. What is Tayor's Theorem?
    • Let Image Upload 11 be analytic in a domain Image Upload 12 and Image Upload 13 is a disk in Image Upload 14. Then 
    • Image Upload 15
  4. What are the Cauchy Riemann equations?
    Let Image Upload 16 be differentiable at Image Upload 17. Then Image Upload 18
  5. What does the Image Upload 19-Inequality say? (Integrals)
    • Let Image Upload 20 be continuous on the contour Image Upload 21 of length Image Upload 22. With Image Upload 23 then 
    • Image Upload 24
  6. Let Image Upload 25 be analytic in the simply connected domain Image Upload 26. If Image Upload 27 is fixed define 

    Image Upload 28.
    • Let Image Upload 29 be any contour interior to Image Upload 30 with starting point Image Upload 31 and terminal point Image Upload 32. Then 
    • Image Upload 33
    • Which is a function independent of choice of contour.
  7. What is Image Upload 34  ?
    Image Upload 35
  8. What is the Maximum Modulus Principle?
    Let Image Upload 36 be analytic on a domain Image Upload 37. If Image Upload 38 is non constant then Image Upload 39 does not attain a maximum on Image Upload 40.
  9. What is a domain? (Complex Analysis)
    An open connected set
  10. What is Gauss' Mean Value Theorem?
    • Let Image Upload 41 be analytic on a simply connected domain Image Upload 42. Let Image Upload 43 then for all Image Upload 44 such that Image Upload 45 
    • Image Upload 46
  11. What are the Cauchy Riemann conditions for differentiability? 
    Let Image Upload 47 be a continuous function. If all the partials of Image Upload 48 exists and satisfies the Cauchy Riemann Equations then Image Upload 49 is differentiable. 
  12. What is the Cauchy-Goursat Theorem?
    • Let Image Upload 50 be analytic on a domain Image Upload 51. Let Image Upload 52 be any simple closed positively oriented curve interior to Image Upload 53. Then
    • Image Upload 54
  13. What is Cauchy's Residue Theorem
    • Let Image Upload 55 be a simple, closed, positively oriented contour. Let Image Upload 56 be analytic on Image Upload 57 and on the interior except at a finite number of points Image Upload 58. Then 
    • Image Upload 59
  14. When is a complex valued function differentiable at Image Upload 60
    Image Upload 61 is differentiable at Image Upload 62 if 

    Image Upload 63 exists.
  15. Define: Analytic at a point
    Image Upload 64 is analytic at Image Upload 65 if there Image Upload 66 exists on a disk around Image Upload 67.
  16. Define: Geometric Series
    • If Image Upload 68 then 
    • Image Upload 69
  17. What are the Taylor Series expansions for sin and cos?
    • Image Upload 70 and 
    • Image Upload 71
  18. If Image Upload 72 what is 
    Image Upload 73
    Image Upload 74
  19. Let Image Upload 75 be a continuous complex valued function defined on D containing the contour Image Upload 76. Let Image Upload 77 be any parameterization of Image Upload 78. Define 
    Image Upload 79
    Image Upload 80
  20. What is the Root Test?
    Let Image Upload 81 be a series satisfying Image Upload 82 then the series converges if Image Upload 83 and diverges if Image Upload 84
  21. What is the residue of Image Upload 85 at Image Upload 86?
    Image Upload 87
    • If Image Upload 88 has a non removable isolated singularity at Image Upload 89 and Image Upload 90 then 
    • Image Upload 91
  22. What is the principle value of the complex logarithm?
    Image Upload 92
  23. What is an isolated singularity?
    Image Upload 93 has an isolated singularity at Image Upload 94 if it is analytic on the the punctured disk Image Upload 95 and not at Image Upload 96
  24. Let Image Upload 97 have a pole of order k at Image Upload 98 compute the residue.
    Image Upload 99
  25. What is Cauchy's Integral Formula
    • Let f be analytic on a simply connected domain D. Let C be any simple, closed, positively oriented contour interior to D. Let Image Upload 100 and Image Upload 101 then 
    • Image Upload 102
  26. Evaluate 
    Image Upload 103
    Using complex analysis
    Substitute Image Upload 104and Image Upload 105 Then integrate on Image Upload 106
  27. What is the Ratio Test?
    Let Image Upload 107 have the property that Image Upload 108 If Image Upload 109 the series converges absolutely and if Image Upload 110 the series diverges.
  28. Explain Deformation of contour
    • Let Image Upload 111 and Image Upload 112 be contours with Image Upload 113 interior to Image Upload 114. if f is analytic on a region that contains both of them and the region between them then 
    • Image Upload 115
  29. When is a function harmonic?
    f is harmonic if it satisfies Laplace's Equation.
  30. What is the Weierstrass M-Test
    Let Image Upload 116 be a series of positive real numbers and Image Upload 117 be a series of complex valued functions defined on T such that Image Upload 118. Then if Image Upload 119 converges so does the power series.
  31. What is a removable singularity
    f has a removable singularity if it has an isolated singularity Image Upload 120, where the Laurent series expansion of f about  Image Upload 121 has no negative powers of Image Upload 122
  32. What is a zero of order k
    f has a zero of order k at Image Upload 123 if Image Upload 124 but Image Upload 125
  33. What is Liouvilles Theorem?
    Entire and bounded means constant.
Card Set
Complex Analysis
A first course in Complex Analysis. Covers, complex functions, differentiation, Integration, Taylor and Laurent Series, Residues.