2003/03/06 Chapter 3 1頁1頁 Chapter 4 : Integration in the Complex Plane 4.1 Introduction to Line Integration.

Slides:

Advertisements

Similar presentations

Chapter 20 Complex variables Cauchy-Riemann relation A function f(z)=u(x,y)+iv(x,y) is differentiable and analytic, there must be particular.

Advertisements

Complex Integration.

MA Day 67 April 22, 2013 Section 13.7: Stokes’s Theorem Section 13.4: Green’s Theorem.

Section 18.4 Path-Dependent Vector Fields and Green’s Theorem.

Stokes Theorem. Recall Green’s Theorem for calculating line integrals Suppose C is a piecewise smooth closed curve that is the boundary of an open region.

Week 5 2. Cauchy’s Integral Theorem (continued)

Chapter 4. Integrals Weiqi Luo (骆伟祺) School of Software

Integration in the Complex Plane CHAPTER 18. Ch18_2 Contents  18.1 Contour Integrals 18.1 Contour Integrals  18.2 Cauchy-Goursat Theorem 18.2 Cauchy-Goursat.

MAT 3730 Complex Variables Section 4.1 Contours

MULTIPLE INTEGRALS MULTIPLE INTEGRALS Recall that it is usually difficult to evaluate single integrals directly from the definition of an integral.

Analytic Continuation: Let f 1 and f 2 be complex analytic functions defined on D 1 and D 2, respectively, with D 1 contained in D 2. If on D 1, then f.

2003/03/26 Chapter 6 1頁1頁 Chapter 6 : Residues & Their Use in Integration 6.1 Definition of the Residues.

2003/02/13 Chapter 4 1頁1頁 Chapter 4 : Multiple Random Variables 4.1 Vector Random Variables.

2003/04/23 Chapter 3 1頁1頁 3.6 Expected Value of Random Variables.

2003/02/14Chapter 2 1頁1頁 Chapter 2 : The Complex Function & Its Derivative 2.1 Introduction.

Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

2006 Fall MATH 100 Lecture 141 MATH 100Class 21 Line Integral independent of path.

2003/04/24 Chapter 5 1頁1頁 Chapter 5 : Sums of Random Variables & Long-Term Averages 5.1 Sums of Random Variables.

Stokes’ Theorem Divergence Theorem

Chapter 16 – Vector Calculus

2006 Fall MATH 100 Lecture 141 MATH 100 Class 20 Line Integral.

© 2010 Pearson Education, Inc. All rights reserved.

Chapter 1, Section 6. Finding the Coordinates of a Midpoint  Midpoint Formula: M( (x1+x2)/2, (y1+y2)/2 )  Endpoints (-3,-2) and (3,4)

Chapter 3 Integral of Complex Function §3.1 Definition and Properties §3.2 Cauchy Integral Theorem §3.3 Cauchy’s Integral Formula §3.4 Analytic and Harmonic.

16 VECTOR CALCULUS.

1 (1) Indefinite Integration (2) Cauchy’s Integral Formula (3) Formulas for the derivatives of an analytic function Section 5 SECTION 5 Complex Integration.

MA Day 55 – April 4, 2013 Section 13.3: The fundamental theorem for line integrals – Review theorems – Finding Potential functions – The Law of.

Chapter 15 Vector Analysis. Copyright © Houghton Mifflin Company. All rights reserved.15-2 Definition of Vector Field.

More on Volumes & Average Function Value Chapter 6.5 March 1, 2007.

CHAPTER Continuity Arc Length Arc Length Formula: If a smooth curve with parametric equations x = f (t), y = g(t), a  t  b, is traversed exactly.

MA Day 53 – April 2, 2013 Section 13.2: Finish Line Integrals Begin 13.3: The fundamental theorem for line integrals.

Riemann Zeta Function and Prime Number Theorem Korea Science Academy Park, Min Jae.

Chapter Integration of substitution and integration by parts of the definite integral.

M M M M 5. Not Listed

SECTION 8 Residue Theory (1) The Residue

Erik Jonsson School of Engineering and Computer Science FEARLESS Engineering ENGR 3300 – 505 Advanced Engineering Mathematics

(MTH 250) Lecture 19 Calculus. Previous Lecture’s Summary Definite integrals Fundamental theorem of calculus Mean value theorem for integrals Fundamental.

ECE 6382 Integration in the Complex Plane David R. Jackson Notes are from D. R. Wilton, Dept. of ECE 1.

The Cauchy–Riemann (CR) Equations. Introduction The Cauchy–Riemann (CR) equations is one of the most fundamental in complex function analysis. This provides.

Integration in Vector Fields

Week 5 2. Cauchy’s Integral Theorem (continued)

Complex Integration  f(z)dz C

Cauchy’s theory for complex integrals

Integration in the Complex Plane

Chapter 9 Vector Calculus.

MTH 324 Lecture # 18 Complex Integration.

1. Complex Variables & Functions

The Residue Theorem and Residue Evaluation

Curl and Divergence.

Chapter 5 Integrals.

Some Theorems Thm. Divergence Theorem

More on Volumes & Average Function Value

The Natural Logarithmic Function: Integration (Section 5-2)

Chapter 1: Introduction

Wednesday, December 05, 2018Wednesday, December 05, 2018

Warm Up Chapter 4.4 The Fundamental Theorem of Calculus

Copyright © 2006 Pearson Education, Inc

Chapter 8 Section 8.2 Applications of Definite Integral

Section 4.2 Contour Integrals

Use Green's Theorem to evaluate the double integral

Presented By Osman Toufiq Ist Year Ist SEM

Chapter 7 Integration.

Evaluate the line integral. {image}

Evaluate the integral. {image}

Warm Up 6.1 Area of a Region Between Two Curves

5.1 Integrals Rita Korsunsky.

Engineering Mathematics

Evaluate the line integral. {image}

Presentation transcript:

2003/03/06 Chapter 3 1頁1頁 Chapter 4 : Integration in the Complex Plane 4.1 Introduction to Line Integration

2003/03/06 Chapter 3 2頁2頁

2003/03/06 Chapter 3 3頁3頁

2003/03/06 Chapter 3 4頁4頁

2003/03/06 Chapter 3 5頁5頁

2003/03/06 Chapter 3 6頁6頁

2003/03/06 Chapter 3 7頁7頁

2003/03/06 Chapter 3 8頁8頁

2003/03/06 Chapter 3 9頁9頁

2003/03/06 Chapter 3 10 頁

2003/03/06 Chapter 3 11 頁

2003/03/06 Chapter 3 12 頁

2003/03/06 Chapter 3 13 頁 4.2 Complex Line Integration

2003/03/06 Chapter 3 14 頁

2003/03/06 Chapter 3 15 頁

2003/03/06 Chapter 3 16 頁

2003/03/06 Chapter 3 17 頁

2003/03/06 Chapter 3 18 頁 ta tb

2003/03/06 Chapter 3 19 頁 Bounds on Line Integrals; the “ML” Inequlity

2003/03/06 Chapter 3 20 頁

2003/03/06 Chapter 3 21 頁

2003/03/06 Chapter 3 22 頁 4.3 Contour Integration and Green’s Theorem Piecewise Smooth Curves = Contours Closed Contour

2003/03/06 Chapter 3 23 頁

2003/03/06 Chapter 3 24 頁

2003/03/06 Chapter 3 25 頁 See Appendix for proofs !

2003/03/06 Chapter 3 26 頁

2003/03/06 Chapter 3 27 頁

2003/03/06 Chapter 3 28 頁

2003/03/06 Chapter 3 29 頁

2003/03/06 Chapter 3 30 頁

2003/03/06 Chapter 3 31 頁

2003/03/06 Chapter 3 32 頁

2003/03/06 Chapter 3 33 頁

2003/03/06 Chapter 3 34 頁

2003/03/06 Chapter 3 35 頁

2003/03/06 Chapter 3 36 頁

2003/03/06 Chapter 3 37 頁

2003/03/06 Chapter 3 38 頁 4.4 Path Independence & Indefinite Integrals

2003/03/06 Chapter 3 39 頁

2003/03/06 Chapter 3 40 頁

2003/03/06 Chapter 3 41 頁

2003/03/06 Chapter 3 42 頁 4.5 Cauchy Integral Formula & Its Extension

2003/03/06 Chapter 3 43 頁

2003/03/06 Chapter 3 44 頁 Choose  ~ 0

2003/03/06 Chapter 3 45 頁 = 0

2003/03/06 Chapter 3 46 頁

2003/03/06 Chapter 3 47 頁

2003/03/06 Chapter 3 48 頁

2003/03/06 Chapter 3 49 頁

2003/03/06 Chapter 3 50 頁