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Chapter 14: Q4P (page 710)

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 = ez

Short Answer

Expert verified

The solutions are, u = ex cos y, v = ey sin y.

The graph is shown in the below image:

Step by step solution

01

To find the solution

The given function is, w = ez

Now, solve the given function by using z = x + iy

w=ez=ex+iy=ex.eiy=excosy+isiny=excosy+iexsiny=u+iv

Hence,u = ex cos y, v = ey sin y , where u is the real function and v is an imaginary function.

02

To sketch the graph of solution

If u =const. , v = const. and for example 0,1,2,3,-1,-2,-3, the graph of the function is given as follows:

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Most popular questions from this chapter

Describe the Riemann surface for .w=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.

e3z-3z-1z4at z = 0

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

  1. If f(x)=u(x)+iυ(x), show that these conditions are equivalent to the equationf*(x)=f(-x) .
  2. Show that

πu(a)=PV∫0∞2xυ(x)x2-a2 dx,      πυ(a)=-PV∫0∞2au(x)x2-a2 dx

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

Find the inverse Laplace transform of the following functions by using (7.16). p+1p(p2+1)

Show that equation (4.4) can be written as (4.5). Then expand each of the fractions in the parenthesis in (4.5) in powers of z and in powers of 1z[see equation (4.7) ] and combine the series to obtain (4.6), (4.8), and (4.2). For each of the following functions find the first few terms of each of the Laurent series about the origin, that is, one series for each annular ring between singular points. Find the residue of each function at the origin. (Warning: To find the residue, you must use the Laurent series which converges near the origin.) Hints: See Problem 2. Use partial fractions as in equations (4.5) and (4.7). Expand a term 1(z-α)in powers of z to get a series convergent for z<α, and in powers of 1z to get a series convergent for z>α.
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