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Chapter 1: Q32P (page 1)

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.

eiz(z2+4) at z = 2i

Short Answer

Expert verified

The residue of the function at z = 2i is R(2i)=3i32e-2

Step by step solution

01

Determine the formula for the higher order poles 

Residue of a function at higher order poles is given as follows:

R[f(z)]=1(n-1)!limz鈫�zkdn-1dzn-1(z-zk)nf(z)

02

Determine the residue of the higher order pole:

Consider the given function as:

f(z)=eiz(z2+4)4

Resolve the function as:

At

The function has a pole of orderatand the residue of higher order pole is given by:

鈥︹��. (1)

03

 Determine the residue of the higher order function as:

Substitute the values in equation (1) and solve as:

Solve further as:

Therefore, the residue of a function at is .

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Most popular questions from this chapter

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

d3dx3(x2ex1-x)at x=0 .

The series 鈭�n-1鈭�1/ns,s>1,is called the Riemann Zeta function,味(s). (In Problem14.2(a)you found味(2)=蟺26. Whennis an even integer, these series can be summed exactly in terms of蟺.) By computer or tables, find

(a)味(4)=鈭�n=1鈭�1n4(b)味(3)=鈭�n=1鈭�1n3(c)味(32)=鈭�n=1鈭�1n3/2

f(x)=ex,a=3.

1+x1-x

Show that the binomial coefficients-1n=(-1)n
See all solutions

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