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Math 132 Section 1 Winter 2009
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Section: 1
Problem Set # 5
Problem ( 1 ) Consider the function Logz (where we use the principal value meaning that the “ar- gument” runs from−π to +π) and let Γ denote the contour which is the quarter circle of radius R in the positive quadrant connecting the real to the imaginary axis. Compute
Γ Logzdz.
Problem ( 2 ) If ˆa denotes a unit vector in the plane, let Dˆa denote the usual directional derivative in the direction of ˆa. [Explicitly, if we write ˆa = (a 1 , a 2 ) and K(x, y) is a two variable function then DˆaK = a 1 Kx + a 2 Ky = ˆa · ∇K] Now let f (z) = u + iv be an analytic function at z. Show that Dˆau = Dˆcv where ˆc is the unit vector rotated 90◦^ from ˆa.
Problem ( 3 ) The Jacobian of a generic map from the (x, y)–plane to the (u, v)–plane: u = u(x, y), v = v(x, y) is defined to be the determinant:
J(x 0 , y 0 ) = det
ux uy vx vy
where the partial derivatives are to be evaluated at the point (x 0 , y 0 ). [It may or may not be recalled that these objects are essential for the consideration of change of variables in the context of multivariable integration.] Show that if u and v are the real and imaginary parts of an analytic function, f (z) = u(x, y) + iv(x, y), then J(x, y) is given by |f ′(z)|^2.
Problem ( 4 ) Suppose that both f (z) and f (z) are analytic (in some region D). Show that this necessarily implies that f is constant.
Problem ( 7 ) The quantity denoted by ii^ is somewhat ambiguous since there are different possible interpretations of how this might be computed. Write down two distinct values for ii^ and justify your perspective(s).
Problem ( 8 ) Using the principle value definition, namely −π < θ ≤ +π, find the value of (1 + i)1+i.
Problem ( 9 ) Find a parameterization for the line segment joining the points z 0 and z 1 in the complex plane. Express your answer in the favored “z(t)” form.
Problem ( 10 ) For z 0 a generic point in the complex plane and a > 0 a positive real number, find a parameterization z = z(t) for the curve (circle) |z − z 0 | = a. Express your parameterization using complex exponentials and, similarly, find ˙z(t).
Problem ( 11 ) Compute the contour integrals
∫
Γ 1
zdz &
Γ 2
zdz
where Γ 1 is the straight line segment from 0 to 1 + i and Γ 2 is the parabolic arc y = x^2 from 0 to 1 + i
Problem ( 12 ) Compute the contour integrals ∫
Γ 1
zdz &
Γ 2
zdz
where Γ 1 is the straight line segment from 0 to 1 + i and Γ 2 is the parabolic arc y = x^2 from 0 to 1 + i
