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دسته بندی دوره ها

Mastering Complex Calculus: From Derivatives to Residues

سرفصل های دوره

Complex Calculus: derivatives of complex variables, contour integration, Laurent series, Fourier series, and residues


1. Introduction to complex functions and derivatives of complex functions
  • 1. functions of a complex variable part 1
  • 2. functions of a complex variable part 2
  • 3. Concept of derivative in complex calculus

  • 2. Integration in Complex Calculus
  • 1. integrals of complex functions and Cauchy theorem
  • 2. Extension of Cauchy theorem
  • 3. Cauchy integral formula part 1
  • 4. Cauchy integral formula part 2

  • 3. Laurent Series, Fourier Series, Taylor Series
  • 1. Laurent series
  • 2. Laurent series in compact form
  • 3. Fourier series derivation from Laurent series
  • 4. Fourier series generalization to any period T
  • 5. Taylor series derivation from Laurent series

  • 4. Residues and Contour Integration
  • 1. Concept of Residue
  • 2. Residue Theorem
  • 3. Calculation of residues and coefficients of the Laurent series
  • 4. Evaluation of a real integral using complex integration (exercise 1)
  • 5. Contour integration to evaluate a real integral (exercise 2)
  • 6. Contour integration to evaluate a real integral (exercise 3)
  • 7. Another integral evaluated using the results of Complex Calculus (exercise 4)
  • 8. Contour integration to evaluate a complex integral (exercise 5)
  • 9. Another contour integration of a real integral - Exercise 6
  • 10. Fresnel integral over the real line (formally derived with the residue theorem)
  • 11. Hilbert transform and its geometric meaning
  • 12. Solution to the diffusion equation using complex calculus and Laplace transform
  • 13. Representation of the Dirac Delta
  • 14. Abel-Plana formula in complex Calculus
  • 15. Convolution of sinc functions using complex calculus

  • 5. How residues aid in the interpretation of the Fourier Transform and its inverse
  • 1. The importance of the Dirac Delta in defining the Inverse Fourier Transform
  • 2. Another integral representation of the Dirac Delta

  • 6. How to use complex calculus to attribute meaning to divergent series
  • 1. Complex calculus to evaluate divergent series
  • 2. Introduction to the analytic continuation of the Riemann zeta function
  • 3. Train of impulses expanded in a Fourier series
  • 4. Poisson summation formula
  • 5. Application of Poisson summation formula
  • 6. Another representation of the Riemann zeta function
  • 7. Functional equation of the Riemann zeta function
  • 8. Evaluation of the Riemann zeta function at s=-3
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    شناسه: 17164
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    مدت زمان: 594 دقیقه
    تاریخ انتشار: ۱۲ مرداد ۱۴۰۲
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