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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:

$\int_{c} f (z) d z = 2 πi \sum_{}^{} R + \Rightarrow \sum_{} R + = \underset{2 πi}{1} \int_{c} f (z) dz$

By definition we know that:

$- \frac{1}{2 πi} \int_{c} f (z) dz = Res (z = \infty)$

02

Add the above equations for the solution

By adding, obtain:

$\sum_{}^{} R^{+} R e s (z = \infty) = 0$

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

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Most popular questions from this chapter

In equation (7.18), let u (x) be an even function and $υ (x)$ be an odd function.

If $f (x) = u (x) + i υ (x)$ , show that these conditions are equivalent to the equation $f^{*} (x) = f (- x)$ .
Show that

$π u (a) = P V \int_{0}^{\infty} \frac{2 x υ (x)}{x^{2} - a^{2}} d x, π υ (a) = - P V \int_{0}^{\infty} \frac{2 a u (x)}{x^{2} - a^{2}} d x$

These are Kramers-Kroning relations. Hint: To find u(a), write the integral for u(a) in (7.18) as an integral from $- \infty$ to 0 plus an integral from 0 to $\infty$ . Then in the to integral $- \infty$ to 0, replace x by -x to get an integral from 0 to $\infty$ , and userole="math" localid="1664350095623" $υ (- x) = - υ (x)$ . Add the two to integrals and simplify. Similarly findrole="math" localid="1664350005594" $υ (a)$ .

Using the definition (2.1) of show that the following familiar formulas hold.

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.

$\frac{e^{2 z} - 1}{z^{2}} at z = 0$

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.

$\frac{z + 2}{(z^{2} + 9) (z^{2} + 1)}$ at $z = 3 i$

For each of the following functions w = f(z) = u +iv, find u and v as functions of x and y. Sketch the graph in (x,y) plane of the images of u = const. and v = const. for several values of and several values of as was done for in Figure 9.3. The curves u = const. should be orthogonal to the curves v = const.

w = e^z

See all solutions

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