Problem sheet 3 Jan 27th 2004

MT290 Complex variable

Ex. 1 Transform each of the following equations into a quadratic equation in e^{z} and hence find all the solutions for z ∈ ℂ to

(a) cosh z = −1; (b) sinh z = \frac{i\sqrt{3}}{2}.

Ex. 2 Determine the principal value of the logarithm of each of the following complex

numbers: (a) −1 − i\sqrt{3}, (b) i − \sqrt{3}, (c) 1 − i, (d) \frac{i − \sqrt{3}}{i − 1}.

Hint: Some of the work has already been done on sheet 1 !

Ex. 3 Find the principal value, and all the other values, of the complex powers:

(a) i^{\frac{2}{π}}, (b) 1^{2i} (c) (i − \sqrt{3})^{i} , (d) (1 − i)^{1+i}.

Ex. 4 Let f(x + iy) = u(x, y) + iv(x, y). Suppose that f is differentiable as a function of a complex variable, and that v(x, y) = (u(x, y))^{2}. Use the Cauchy-Riemann equations to show that

\frac{∂u}{∂x} = −4u^{2} \frac{∂u}{∂x}

and thus deduce that f(z) is constant. (Note: 1 + 4u^{2} ≠ 0 !)

Ex. 5 By writing tan z in terms of w = e^{iz} write down the quadratic equation w must satisfy if

tan z = a.

Hence find z if tan z = \sqrt{3} − 2i. By considering the quadratic equation which w satisfies, and using the fact that e^{z} = 0 has no solution, show that tan z takes all values in ℂ with two exceptions.