Complex Analysis - Maths KU

Chapter 7 Review Quiz 449
13. If w = f(z) =u(x, y)+iv(x, y) is a conformal mapping of a domain D onto
the upper half-plane v>0 and if Φ(u, v) is a harmonic function for v>0,
then φ(x, y) =Φ(u(x, y), v(x, y)) is harmonic on D.
14. If ψ(x, y) is a function defined on a domain D and if the boundary of D is
a level curve of ψ(x, y), then ψ(x, y) is the stream function of an ideal fluid
in D.
15. Given a domain D, there can be more than one flow of an ideal fluid that
remains inside of D.
In Problems 16–30, try to fill in the blanks without referring back to the text.
16. The analytic function f(z) = cosh z is conformal except at z = .
17. Conformal mappings preserve both the magnitude and the of an
angle.
18. The mapping is an example of a mapping that is conformal at every
point in the complex plane.
19. If f ′ (z0) =f ′′ (z0) = 0 and f ′′′ (z0) �= 0, then the mapping w = f(z)
the magnitude of angles at the point z0.
20. T (z) = is a linear fractional transformation that maps the points 0,
1+i, and i onto the points 1, i, and ∞.
21. The image of the circle |z − 1| = 2 under the linear fractional transformation
T (z) =(2z − i)/ (iz +1) is a .
22. The image of a line L under the linear fractional transformation
T (z) =(iz − 2)/ (3z +1− i) is a circle if and only if the point z =
is on L.
23. The cross-ratio of the points z, z1, z2, and z3 is and .
24. The derivative of a Schwarz-Christoffel mapping from the upper half-plane onto
the triangle with vertices at 0, 1, and 1 + i is f ′ (z) = .
25. If f ′ (z) =A(z +1) −1/2 z −1/4 , then w = f(z) maps the upper half-plane onto a
polygonal region with interior angles .
26. The Poisson integral formula gives an integral solution φ(x, y) to a Dirichlet
problem in the upper half-plane y>0 provided the function f(x) =φ(x, 0) is
and on −∞