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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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