Math Problems Vault

Problem 30.1

Find functions $w(z)$ that conformally map the half-plane $\left\{ z: \mathrm{Im}\left(z\right) > 0\right\}$ onto the polygons depicted in Fig. 30.5-30.18, in such a way that $w(0) = A_{1}, w(1) = A_{2}, w(\infty ) = A_{3}$ (the vertices $A_{k}$ are marked in the figures).

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(1)

Fig. 30.5

(2)

Fig. 30.6

(3)

Fig. 30.7

(4)

Fig. 30.8

(5)

Fig. 30.9

(6)

Fig. 30.10

(7)

Fig. 30.11

(8)

Fig. 30.12

(9)

Fig. 30.13

(10)

Fig. 30.14

(11)

Fig. 30.15

(12)

Fig. 30.16

(13)

Fig. 30.17

(14)

Fig. 30.18

Problem 30.2

Find some functions $w(z)$ that conformally map the half-plane $\left\{ z: \mathrm{Im}\left(z\right) > 0\right\}$ onto the symmetric polygons depicted in Fig. 30.19-30.24.

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(1)

Fig. 30.19

(2)

Fig. 30.20

(3)

Fig. 30.21

(4)

Fig. 30.22

(5)

Fig. 30.23

(6)

Fig. 30.24

Problem 30.3

(1)(2)(3)

Find functions $w(z)$ that conformally map the region $\left\{ z:\left|z\right| > 1\right\}$ onto the polygons with interior point $w = \infty$ depicted in Fig. 30.25-30.27, in such a way that $w(\infty ) = \infty , w(1) = A_{1}$ .

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(1)

Fig. 30.25

(2)

Fig. 30.26

(3)

Fig. 30.27

Problem 30.4

(1)(2)(3)(4)

Find functions $w(z)$ that conformally map the half-plane $\left\{ z: \mathrm{Im}\left(z\right) > 0\right\}$ onto the regions depicted in Fig. 30.28-30.31, in such a way that

w(0) = A_{1}, \quad w(1) = A_{2}, \quad w(\infty ) = A_{3}.

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(1)

Fig. 30.28

(2)

Fig. 30.29

(3)

Fig. 30.30

(4)

Fig. 30.31

Problem 30.5

(1)(2)(3)

Find functions $w(z)$ that conformally map the entire $z$ -plane cut along the segment $[-1,1]$ onto the regions depicted in Fig. 30.32-30.34, in such a way that $w(\infty ) = \infty , w(1) = A_{1}$ .

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(1)

Fig. 30.32

(2)

Fig. 30.33

(3)

Fig. 30.34

Problem 30.6

(1)(2)(3)

Find functions $w(z)$ that conformally map the strip $\left\{ z:\left|\operatorname {Im}\right| < \frac{1}{2}\right\}$ onto the regions depicted in Fig. 30.35-30.37, in such a way that $w(0) = 0, w(+\infty ) = A_{1}$ .

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(1)

Fig. 30.35

(2)

Fig. 30.36

(3)

Fig. 30.37

Problem 30.7

(1)(2)

Find functions $w(z)$ that conformally map the half-strip $\left\{ z: \mathrm{Re}\left(z\right) > 0,\left|\mathrm{Im}\left(z\right)\right| < \frac{\pi }{2}\right\}$ onto the regions depicted in Fig. 30.38-30.39, in such a way that

w\left(-\frac{\pi i}{2}\right) = A_{1}, \quad w(\infty ) = A_{2}, \quad w\left(\frac{\pi i}{2}\right) = A_{3}

?

(1)

Fig. 30.38

(2)

Fig. 30.39

Problem 30.8

Find some function $w(z)$ that conformally maps the half-plane $\left\{ z: \mathrm{Im}\left(z\right) > 0\right\}$ onto the $w$ -plane with cuts along the rays

[k i, k i+i \infty ], \quad \left[\left(k+\frac{1}{4}\right) i,\left(k+\frac{1}{4}\right) i+i \infty \right] \quad (k = 0, \pm 1, \pm 2, \ldots )

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Problem 30.9

Find some function $w(z)$ that conformally maps the half-plane $\left\{ z: \mathrm{Im}\left(z\right) > 0\right\}$ onto the half-plane $w$ from which the strips are removed

\left\{ w: \mathrm{Re}\left(w\right) < 0,\left|k-\mathrm{Im}\left(w\right)\right| < \frac{1}{4}\right\} \quad (k = 0, \pm 1, \pm 2, \ldots )

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Problem 30.10

(1)(2)(3)

Find functions $w(z)$ that conformally map the half-plane $\left\{ z: \mathrm{Im}\left(z\right) > 0\right\}$ onto the polygons depicted in Fig. 30.40-30.42, in such a way that $w(0) = A_{1}, w(1) = A_{2}, w(\infty ) = A_{3}$ .

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(1)

Fig. 30.40

(2)

Fig. 30.41

(3)

Fig. 30.42

Problem 30.11

Find a function $w(z)$ that conformally maps the disk $\left\{ z:\left|z\right| < 1\right\}$ onto a regular $n$ -gon inscribed in the disk $\left\{ w:\left|w\right| < 1\right\}$ , in such a way that $w(0) = 0, w(1) = 1$ .

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