Question 8.22: Let C be a curve in the z plane with parametric equations x ...

Schaum’s Outline of Complex Variables

Let $C$ be a curve in the $z$ plane with parametric equations $x=F(t), y=G(t)$ . Show that the transformation

$z=F(w)+i G(w)$

maps the real axis of the $w$ plane onto $C$ .

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

Related Answered Questions

Question: 8.4

Prove that the angle between two curves C1 and C2 passing through the point z0 in the z plane [see Figs. 8-1 and 8-2, page 243] is preserved [in magnitude and sense] under the transformation w = f(z), i.e., the mapping is conformal, if f(z) is analytic at z0 and f′0(z0)≠0. ...

Verified Answer:

By Problem 8.3, each curve is rotated through the ...

Question: 8.28

Suppose the circle C of Problem 8.27 is moved so that its center is in the upper half plane but that it still passes through z = 1 and encloses z = -1. Determine the image of C under the transformation w = 1/2 (z + 1/z). ...

Verified Answer:

As in Problem 8.27, since

z=1

is a ...

Question: 8.25

(a) Find a function which maps the upper half of the z plane of Fig. 8-85 onto the shaded region in the w plane of Fig. 8-84. (b) Discuss the case where b → 0. ...

Verified Answer:

(a) The interior angles at

Q

and [l...

Question: 8.24

Find a function that maps the upper half of the z plane onto the interior of a triangle in the w plane [Fig. 8-82]. ...

Verified Answer:

Consider the upper half of the

z

pl...

Question: 8.23

Find a transformation that maps the real axis in the w plane onto the ellipse (x²/a²) + (y²/b²) = 1 in the z plane. ...

Verified Answer:

A set of parametric equations for the ellipse is g...

Question: 8.21

Find a transformation that maps the unit circle in the ζ plane onto a polygon in the w plane. ...

Verified Answer:

The

x

axis in the

z

...

Question: 8.20

Determine a function that maps the upper half of the z plane onto each of the indicated regions in the w plane. ...

Verified Answer:

(a) Let points

P, Q, S

, and

...

Question: 8.19

Suppose in the Schwarz –Christoffel transformation (8.9) or (8.10), page 247, one point, say xn, is chosen at infinity. Show that the last factor is not present. ...

Verified Answer:

In (8.9), page 247, let

A=K /\left(-x_{n}\r...

Question: 8.18

Prove that for closed polygons, the sum of the exponents (a1/π) – 1, (a2/π) 1, … , (an/p) – 1 in the Schwarz –Christoffel transformation (8.9) or (8.10), page 247, is equal to -2. ...

Verified Answer:

The sum of the exterior angles of any closed polyg...

Question: 8.17

Establish the validity of the Schwarz-Christoffel transformation. ...

Verified Answer:

We must show that the mapping function obtained fr...