Complex Analysis

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Author:
NhanNguyen
ID:
225152
Filename:
Complex Analysis
Updated:
2013-06-26 22:07:29
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MATH Complex Analysis
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Description:
A first course in Complex Analysis. Covers, complex functions, differentiation, Integration, Taylor and Laurent Series, Residues.
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  1. What is the Comparison Test?
    Let  be a convergent series of real number.  for all . If | then  converges absolutely.
  2. What is Morera's Theorem? 
    (Converse of Cauchy- Goursat)
    Let  be continuous on a domain  if  for all simple closed contours in  then  is analytic on D
  3. What is Tayor's Theorem?
    • Let  be analytic in a domain  and  is a disk in . Then 
  4. What are the Cauchy Riemann equations?
    Let  be differentiable at . Then 
  5. What does the -Inequality say? (Integrals)
    • Let  be continuous on the contour  of length . With  then 
  6. Let  be analytic in the simply connected domain . If  is fixed define 

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

     exists.
  15. Define: Analytic at a point
     is analytic at  if there  exists on a disk around .
  16. Define: Geometric Series
    • If  then 
  17. What are the Taylor Series expansions for sin and cos?
    •  and 
  18. If  what is 
  19. Let  be a continuous complex valued function defined on D containing the contour . Let  be any parameterization of . Define 
  20. What is the Root Test?
    Let  be a series satisfying  then the series converges if  and diverges if 
  21. What is the residue of  at ?
    • If  has a non removable isolated singularity at  and  then 
  22. What is the principle value of the complex logarithm?
  23. What is an isolated singularity?
     has an isolated singularity at  if it is analytic on the the punctured disk  and not at 
  24. Let  have a pole of order k at  compute the residue.
  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  and  then 
  26. Evaluate 

    Using complex analysis
    Substitute and  Then integrate on 
  27. What is the Ratio Test?
    Let  have the property that  If  the series converges absolutely and if  the series diverges.
  28. Explain Deformation of contour
    • Let  and  be contours with  interior to . if f is analytic on a region that contains both of them and the region between them then 
  29. When is a function harmonic?
    f is harmonic if it satisfies Laplace's Equation.
  30. What is the Weierstrass M-Test
    Let  be a series of positive real numbers and  be a series of complex valued functions defined on T such that . Then if  converges so does the power series.
  31. What is a removable singularity
    f has a removable singularity if it has an isolated singularity , where the Laurent series expansion of f about   has no negative powers of 
  32. What is a zero of order k
    f has a zero of order k at  if  but 
  33. What is Liouvilles Theorem?
    Entire and bounded means constant.

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