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Math 132 Section 1 Winter 2009
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Section: 1
Problem Set # 3
Problem ( 1 ) In many situations, one encounters coupled differential equations; often times complex analysis can be quite useful. A famous example is the system dA dt
= B(t);
dB dt
= −A(t).
Here A(t) and B(t) are unknown functions of t; all quantities are real. Now consider the complex valued function (of the single real variable t)
Q(t) = A(t) + iB(t)
Note that by adding i×[left equation] to the right equation, you get a straightforward complex equation for Q(t). Using this method, solve this above system subject to the initial condition A(t =
- = A 0 , B(t = 0) = B 0.
Problem ( 2 ) Write the function f (z) in the form u(x, y) + iv(x, y) – u and v real – where
f (z) = 4z^2 + 6z + i + 1.
Problem ( 3 ) Write the function f (z) in the form u(x, y) + iv(x, y) – u and v real – where
f (z) =
z
Problem ( 4 ) Write the function f (z) in the form u(x, y) + iv(x, y) – u and v real – where
f (z) =
2 z^2 + 3 |z − 1 |
Problem ( 5 ) Show directly that for any z 6 = 1,
1 + z + z^2 + · · · + zn^ =
1 − zn+ 1 − z
Problem ( 8 ) Find all complex numbers z for which (^2) 2+−zz is pure real.
Problem ( 9 ) Find all complex numbers z for which (^2) 2+−zz is pure imaginary.
Problem ( 10 ) Consider the most general linear complex function
f (z) = P x + Qy
where P and Q denote arbitrary complex constants. Find the conditions on P and Q for which f has a complex derivative.
Problem ( 11 ) Suppose f (z) = u(x, y) + iv(x, y) is expressed in polar coordinates:f (z) = A(r, θ) + iB(r, θ). Derive the polar Cauchy–Riemann equations satisfied by A and B if u and v obey the usual (Cartesian) CR equations.
Problem ( 14 ) For any complex number c = a + ib (a and b real), write down the real and imaginary parts of the function f (z) = ecz^ and show that the derivative exists and that in fact, f ′(z) = cecz
[You need not prove the existence of the derivative from first principles, you may use the fact that the CR equations are necessary and sufficient.]
Problem ( 15 ) Let f (z) = u(x, y) + iv(x, y) be a complex function which is everywhere differentiable
- so, explicitly the CauchyRiemann equations are satisfied. Now consider the two level curves
u(x, y) = c 1 & v(x, y) = c 2
where c 1 and c 2 are constants (whose particular value plays no role). It is supposed that the two curves intersect at some point (x 0 , y 0 ). Show that they do so orthogonally; that is to say at the point of intersection, the tangents to the curves are at right angles.
Problem ( 16 ) Let f (z) = u(x, y) + iv(x, y) and g(z) = p(x, y) + iq(x, y) denote two functions that both have a complex derivative at z = z 0 = x 0 + iy 0. Consider the product f (z)g(z). Show, by establishing the Cauchy–Riemann equations for f g, that
(i) The product f g has a complex derivative at z = z 0
(ii) The derivative of f g obeys the usual product rule.
