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MA231 Vector Analysis
Example Sheet 4
2010, term 1 Stefan Adams
Hand in solutions to questions B1, B2, B3 and B4 by 3pm Monday of week 10.
On this example sheet, all contour integrals around simple closed curves are taken in anti- clockwise direction.
A1 Cauchy-Riemann equations
(a) Find the real and imaginary parts of f (z) = ez^ and show they satisfy the Cauchy Riemann equations. (b) Suppose that a holomorphic function f : C → C satisfies Re(f (z)) = 0 for all z ∈ C. Show that f is a constant. (c) Suppose that f (x + iy) = u(x, y) + iv(x, y) is holomorphic on C. Show that ∆u = ∆v = 0 on R^2 (you may assume that the second partial derivatives exist and are continuous).
A2 Complex contour integrals
(a) Calculate
γ f^ dz^ when^ f^ (x^ +^ iy) =^ xy^ and^ γ(t) = e
it (^) for t ∈ [0, π].
(b) Calculate
C f^ dz^ where^ f^ (z) =^ z^ and^ C^ is the straight line from^0 to^ 1 +^ i. A3 Contour integrals via Cauchy’s integral representation
(a) Using Cauchy’s integral formula calculate the following contour integrals:
(i)
∂B(0,2)
ez z − 1 dz, (ii)
∂B(0,2)
ez πi − 2 z dz, (iii)
∂B(0,2)
ez 6 πi − 2 z dz.
(b) Using Cauchy’s integral formula for the coefficients ck of a Taylor series, evaluate the contour integral ∫
∂B(i,2)
ez (z − 1)n^ dz when n is a positive integer.
A4 More contour integrals Evaluate
∂B
1 1+z^2 dz^ for each of the following balls: (i) B(1, 1), (ii) B(i, 1), (iii) B(−i, 1), (iv) B(0, 2), (v) B(3i, π).
A5 Evaluation of a real integral Using a contour integral of the function
f (z) =
eπiz z^2 − 2 z + 2 around a semicircular contour, evaluate the real integrals ∫ (^) ∞
−∞
cos(πx) x^2 − 2 x + 2
dx and
−∞
sin(πx) x^2 − 2 x + 2
dx.
B1 Cauchy-Riemann equations
(a) Suppose that a holomorphic function f is of the special form f (x + iy) = u(y) + iv(x). Show that f (z) = aiz + b for some a ∈ R and b ∈ C. (b) At what points z ∈ C are the functions f (z) = zz^2 and g(z) = g(x + iy) = (x − xy^2 ) + ix^2 y differentiable? At what points are f and g holomorphic? (c) Let a, b ∈ R be real numbers and fa,b : C → C, fa,b(z) = fa,b(x + iy) = cos x(cosh y + a sinh y) + i sin x(cosh y + b sinh y). Determine the real num- bers a, b ∈ R for which the function fa,b is complex differentiable on C. Write then the function fa,b as a function of the complex variable z.
B2 Contour integrals
(a) Calculate the integral
C f^ dz^ where^ f^ (z) =^ z^ and^ C^ is the boundary of the triangle with vertices 1 , i, − 1 , traversed in an anticlockwise direction. (b) For ε > 0 let γε be the piece of circular arc of radius ε parameterised by γε(t) = εeit^ for t ∈ [α, β] ⊆ [0, 2 π]. Show, for continuous f : C → C, that ∫
γε
f (z) z
dz → f (0)(β − α)i as ε → 0.
(c) Evaluate the following integrals anticlockwise around the boundary of the ball B(0, 2):
(i)
z^5 + 3 z − i dz, (ii)
ez^2 (z − 1)^3 dz.
B3 More contour integrals
(a) Each of the following integrals is zero. Give a brief reason for each example.
(i)
∂B(1,2)
sin z z
dz, (ii)
∂B(1,2)
z + 2
dz, (iii)
∂B(1,2)
(z − 2)^3
dz.
(b) Let C = {z = x + iy ∈ C : x
2 a +^
y^2 b = 1}^ be the ellipse with main axes^ a^ and^ b parallel to the x- axis respectively y-axis. Calculate in dependence on a 6 = 1 and b 6 = π 2 the following contour integrals (along C in positive direction)
(i)
C
e−z (z − i π 2 )^2
dz, (ii)
C
z^3 − 4 z^2 + sin z (z − 1)^3
dz.
B4 Evaluating a real integral The aim is to evaluate
−∞
sin^2 x x^2 dx. The strategy is to do a contour integral of the function f (z) = e 2 iz (^) − 1 z^2 +a^2 for real^ a >^0 around the contour^ γ^ consisting of two parts: γ 1 the semi-circle
z ∈ C
|z| = R, Im(z) > 0
, and γ 2 the straight line segment from −R to R. (i) Explain why
γ 1 f^ (z) dz^ →^0 as^ R^ → ∞. (ii) Use Cauchy’s integral formula to show that
γ f^ (z) dz^ =^
π a
e−^2 a^ − 1
(iii) Show that e^2 ix^ − 1 = −2 sin^2 x + i sin 2x and hence that
γ 2 f^ (z) dz^ is real. (iv) Evaluate
−∞
sin^2 x x^2 +a^2 dx. (v) Let a → 0 to find
−∞
sin^2 x x^2 dx.
