MTH 324 Lecture 31 Uniform Convergence 1 Previous

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MTH 324 Lecture # 31 Uniform Convergence 1

MTH 324 Lecture # 31 Uniform Convergence 1

Previous Lecture’s Review • Fourier integrals • Indented integrals 2

Previous Lecture’s Review • Fourier integrals • Indented integrals 2

Lecture’s Outline • Point wise convergence of real sequences • Uniform convergence of real

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

Point wise convergence of real sequence: 4

Example: Solution: 5

Example: Solution: 5

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Uniform convergence of real sequence: 7

Uniform convergence of real sequence: 7

Example: Solution: 8

Example: Solution: 8

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Cauchy’s general principle of convergence: 10

Cauchy’s general principle of convergence: 10

Uniform convergence and continuity: Uniform convergence and Integration: 11

Uniform convergence and continuity: Uniform convergence and Integration: 11

Convergence and uniform convergence of series of functions: 12

Convergence and uniform convergence of series of functions: 12

Weierstrass’s M-test for uniform convergence of real series: 13

Weierstrass’s M-test for uniform convergence of real series: 13

Example: Solution: 14

Example: Solution: 14

Uniform convergence and integration: 15

Uniform convergence and integration: 15

Point wise convergence of complex sequence of functions: 16

Point wise convergence of complex sequence of functions: 16

Remark: 17

Remark: 17

Example: Solution: 18

Example: Solution: 18

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Theorem: Proof: 20

Theorem: Proof: 20

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Theorem: Proof: 23

Theorem: Proof: 23

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Uniform convergence of series of complex functions: 25

Uniform convergence of series of complex functions: 25

Weierstrass’s M-test for uniform convergence of complex series: 26

Weierstrass’s M-test for uniform convergence of complex series: 26

Example: Solution: 27

Example: Solution: 27

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References • Complex variables and applications by James Brown and Ruel Churchill • Introduction

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