Lecture’s Outline • Point wise convergence of real sequences • Uniform convergence of real sequence and series • Point wise convergence of complex sequences • Uniform convergence of real sequence and series 3
Point wise convergence of real sequence: 4
Example: Solution: 5
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Uniform convergence of real sequence: 7
Example: Solution: 8
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Cauchy’s general principle of convergence: 10
Uniform convergence and continuity: Uniform convergence and Integration: 11
Convergence and uniform convergence of series of functions: 12
Weierstrass’s M-test for uniform convergence of real series: 13
Example: Solution: 14
Uniform convergence and integration: 15
Point wise convergence of complex sequence of functions: 16
Remark: 17
Example: Solution: 18
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Theorem: Proof: 20
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Theorem: Proof: 23
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Uniform convergence of series of complex functions: 25
Weierstrass’s M-test for uniform convergence of complex series: 26
Example: Solution: 27
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References • Complex variables and applications by James Brown and Ruel Churchill • Introduction to Real Analysis by B. S. Vatsa • Introduction to Complex Analysis by w w L Chen 30