/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q16P To prove that the sum of the res... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Q16P (page 705)

To prove that the sum of the residues at finite points plus the residence at infinity is zero.

Short Answer

Expert verified

Sum of the residues at the singularity is zero.

Step by step solution

01

Use residue theorem

Proof: - Let C be a closed contour which encloses all the singularities of f(z) except that at infinity, then by residue theorem as follows:

鈭�cf(z)dz=2蟺颈鈭�R+鈬�鈭�R+=12蟺颈鈭�cf(z)dz

By definition we know that:

-12蟺颈鈭�cf(z)dz=Res(z=鈭�)

02

Add the above equations for the solution

By adding, obtain:

鈭�R+Res(z=鈭�)=0

That is, sum of the residues at the singularity is zero.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Using the definition (2.1) of , show that the following familiar formulas hold. Hint : Use the same methods as for functions of a real variable.

28.. (See hint below.)

Problem 28 is the chin rule for the derivative of a function of a function.

(a) Show that if f(z)tends to a finite limit as z tends to infinity, then the residue of f(z) at infinity is.

(b) Also show that iff(z)tends to zero as z tends to infinity, then the residue of f(z) at infinity is -limz2f'(z).

Find the real and imaginary parts u(x,y) andv(x,y) of the following functions.

z

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

e2蟺颈z1-z3at z=e2蟺颈3

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer.

e2z4coshz-5atz=ln2
See all solutions

Recommended explanations on Physics Textbooks

Energy Physics

Read Explanation

Magnetism

Read Explanation

Waves Physics

Read Explanation

Linear Momentum

Read Explanation

Torque and Rotational Motion

Read Explanation

Translational Dynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.