Complex Analysis

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  1. What is the Comparison Test?
    Let Image Upload be a convergent series of real number. Image Upload for all Image Upload. If |Image Upload then Image Upload converges absolutely.
  2. What is Morera's Theorem? 
    (Converse of Cauchy- Goursat)
    Let Image Upload be continuous on a domain Image Upload if Image Upload for all simple closed contours in Image Upload then Image Upload is analytic on D
  3. What is Tayor's Theorem?
    • Let Image Upload be analytic in a domain Image Upload and Image Upload is a disk in Image Upload. Then 
    • Image Upload
  4. What are the Cauchy Riemann equations?
    Let Image Upload be differentiable at Image Upload. Then Image Upload
  5. What does the Image Upload-Inequality say? (Integrals)
    • Let Image Upload be continuous on the contour Image Upload of length Image Upload. With Image Upload then 
    • Image Upload
  6. Let Image Upload be analytic in the simply connected domain Image Upload. If Image Upload is fixed define 

    Image Upload.
    • Let Image Upload be any contour interior to Image Upload with starting point Image Upload and terminal point Image Upload. Then 
    • Image Upload
    • Which is a function independent of choice of contour.
  7. What is Image Upload  ?
    Image Upload
  8. What is the Maximum Modulus Principle?
    Let Image Upload be analytic on a domain Image Upload. If Image Upload is non constant then Image Upload does not attain a maximum on Image Upload.
  9. What is a domain? (Complex Analysis)
    An open connected set
  10. What is Gauss' Mean Value Theorem?
    • Let Image Upload be analytic on a simply connected domain Image Upload. Let Image Upload then for all Image Upload such that Image Upload 
    • Image Upload
  11. What are the Cauchy Riemann conditions for differentiability? 
    Let Image Upload be a continuous function. If all the partials of Image Upload exists and satisfies the Cauchy Riemann Equations then Image Upload is differentiable. 
  12. What is the Cauchy-Goursat Theorem?
    • Let Image Upload be analytic on a domain Image Upload. Let Image Upload be any simple closed positively oriented curve interior to Image Upload. Then
    • Image Upload
  13. What is Cauchy's Residue Theorem
    • Let Image Upload be a simple, closed, positively oriented contour. Let Image Upload be analytic on Image Upload and on the interior except at a finite number of points Image Upload. Then 
    • Image Upload
  14. When is a complex valued function differentiable at Image Upload
    Image Upload is differentiable at Image Upload if 

    Image Upload exists.
  15. Define: Analytic at a point
    Image Upload is analytic at Image Upload if there Image Upload exists on a disk around Image Upload.
  16. Define: Geometric Series
    • If Image Upload then 
    • Image Upload
  17. What are the Taylor Series expansions for sin and cos?
    • Image Upload and 
    • Image Upload
  18. If Image Upload what is 
    Image Upload
    Image Upload
  19. Let Image Upload be a continuous complex valued function defined on D containing the contour Image Upload. Let Image Upload be any parameterization of Image Upload. Define 
    Image Upload
    Image Upload
  20. What is the Root Test?
    Let Image Upload be a series satisfying Image Upload then the series converges if Image Upload and diverges if Image Upload
  21. What is the residue of Image Upload at Image Upload?
    Image Upload
    • If Image Upload has a non removable isolated singularity at Image Upload and Image Upload then 
    • Image Upload
  22. What is the principle value of the complex logarithm?
    Image Upload
  23. What is an isolated singularity?
    Image Upload has an isolated singularity at Image Upload if it is analytic on the the punctured disk Image Upload and not at Image Upload
  24. Let Image Upload have a pole of order k at Image Upload compute the residue.
    Image Upload
  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 and Image Upload then 
    • Image Upload
  26. Evaluate 
    Image Upload
    Using complex analysis
    Substitute Image Uploadand Image Upload Then integrate on Image Upload
  27. What is the Ratio Test?
    Let Image Upload have the property that Image Upload If Image Upload the series converges absolutely and if Image Upload the series diverges.
  28. Explain Deformation of contour
    • Let Image Upload and Image Upload be contours with Image Upload interior to Image Upload. if f is analytic on a region that contains both of them and the region between them then 
    • Image Upload
  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 be a series of positive real numbers and Image Upload be a series of complex valued functions defined on T such that Image Upload. Then if Image Upload 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, where the Laurent series expansion of f about  Image Upload has no negative powers of Image Upload
  32. What is a zero of order k
    f has a zero of order k at Image Upload if Image Upload but Image Upload
  33. What is Liouvilles Theorem?
    Entire and bounded means constant.
Card Set:
Complex Analysis
2013-06-27 02:07:29
MATH Complex Analysis

A first course in Complex Analysis. Covers, complex functions, differentiation, Integration, Taylor and Laurent Series, Residues.
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