Analytic Functions|| Examples of Cauchy's Riemann(C.R) Equations|| Complex Analysis

COURSE: COMPLEX ANALYSIS
CHAPTER # 2 Analytic Functions.
Topic: In this lecture I shall discuss some important examples related to Cauchy's Riemann(C.R) Equations
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   • COMPLEX ANALYSIS|| HIGHER MATHMATICS|...  
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FOLLOWING ARE THE CONTENTS OF COMPLEX ANALYSIS
1. Complex numbers, complex planes, complex functions
2. Analytic functions
3. Entire functions
4. Harmonic functions
5. Elementary functions: complex exponential, logarithmic and hyperbolic functions
6. Infinite Series
7. Power series, derived series, radius of convergence
8. Taylor series and Laurent series
9. Conformal Representation
10. Transformation, conformal transformation
11. Linear transformation
12. Möbius transformations
13. Complex Integration
14. Complex integrals
15. Cauchy-Goursat theorem
16. Cauchy’s integral formula and their consequences
17. Liouville’s theorem
18. Morera’s theorem
19. Derivative of an analytic function
Singularity and Poles
 Review of Laurent series
 Zeros, singularities
 Poles and residues
Contour Integration
 Cauchy’s residue theorem
 Applications of Cauchy’s residue theorem
Expansion of Functions and Analytic Continuation
 Mittag-Leffler theorem
 Weierstrass’s factorization theorem
 Analytic continuation
Elliptic Functions
 Periodic functions
 Elliptic functions and its properties
 Weierstrass function  (z)
 Differential equation satisfied by  (z)
 Integral formula for  (z)
 Addition theorem for  (z)
 Duplication formula for  (z)
 Elliptic functions in terms of Weierstrass function with the same periods
 Quasi periodic functions: The zeta and sigma functions of Weierstrass
 Jacobian elliptic functions and its properties
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