A First Course in Complex Analysis by Dennis Zill | 9780763714376 | 0763714372 | Holooly

Structural Analysis

A First Course in Complex Analysis

146 SOLVED PROBLEMS

Question: 2.3.3

Image of a Circle under Magnification. Find the image of the circle C given by |z| = 2 under the linear mapping M(z) = 3z. ...

Verified Answer:

Since M is a magnification with magnification fact...

Question: 2.3.4

Image of a Rectangle under a Linear Mapping. Find the image of the rectangle with vertices −1+i, 1+i, 1+2i, and −1+2i under the linear mapping f(z) = 4iz + 2 + 3i. ...

Verified Answer:

Let

S

be the rectangle with the giv...

Question: 2.4.1

Image of a Circular Arc under w = z². Find the image of the circular arc defined by |z| = 2, 0 ≤ arg(z) ≤ π/2, under the mapping w = z². ...

Verified Answer:

Let C be the circular arc defined by |z| = 2, 0 ≤ ...

Question: 2.4.3

Image of a Triangle under w = z². Find the image of the triangle with vertices 0, 1 + i, and 1 − i under the mapping w = z². ...

Verified Answer:

Let S denote the triangle with vertices at 0, 1 + ...

Question: 2.4.4

Image of a Circular Wedge under w = z³. Determine the image of the quarter disk defined by the inequalities |z| ≤ 2, 0 ≤ arg(z) ≤ π/2, under the mapping w = z³. ...

Verified Answer:

Let S denote the quarter disk and let

S^{\p...

Question: 2.4.8

Image of a Circular Sector under w = z½. Find the image of the set S defined by |z| ≤ 3, π/2 ≤ arg(z) ≤ 3π/4, under the principal square root function. ...

Verified Answer:

Let

S^{\prime}

denote the image of ...

Question: 7.3.4

Using the Schwarz-Christoffel Formula. Use the Schwarz-Christoffel formula (6) to construct a conformal mapping from the upper half-plane onto the nonpolygonal region defined by v ≥ 0, with the horizontal half-line v = π, −∞ < u ≤ 0, deleted. See Figure 7.26(c). ...

Verified Answer:

Let

u_{0}

be a point on the nonposi...

Question: 7.5.1

A Heat Flow Application. Determine the steady-state temperature φ in the domain D consisting of allpoints outside of the two circles |z| = 1 and|z-5/2| = 1/2, shown in color in Figure 7.48(a), that satisfies the indicated boundary conditions. ...

Verified Answer:

The steady-state temperature φ is a solution of La...

Question: 7.5.2

An Electrostatics Application. Determine the electrostatic potential φ in the domain D between the circles |z| = 1 and|z-1/2|= 1/2 , shown in color in Figure 7.50(a), that satisfies the indicated boundary conditions. ...

Verified Answer:

The electrostatic potential

\boldsymbol{\ph...

Question: 7.5.6

Streamlines Defined Parametrically. Construct a flow of an ideal fluid in the domain D consisting of all points in the upper half-plane y > 0 excluding the points on the ray y = π, −∞ < x ≤ 0, shown in Figure 7.58. ...

Verified Answer:

In Example 4 of Section 7.3 we used the Schwarz-Ch...

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