By Leif Mejlbro
ISBN-10: 8776813843
ISBN-13: 9788776813840
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Extra resources for Complex Functions Examples c-2 – Analytic Functions
Example text
G. the parametric description z = eiθ , θ∈ − π π , , 2 2 hence by insertion, C |z| dz = π 2 −π 2 i eiθ dθ = eiθ π 2 = i − (−i) = 2i. 1 Clearly, the value of the line integral depends in this case of the path of integration. ♦ 1 dz, where C denotes the curve with the parametric description z π . 5 Compute the line integral z = 1 to z = i. C iπ . 2 x2 + i y 3 dz, where C is the straight line segment from A parametric description of C is given by x(t) = 1 − t, y(t) = t, t ∈ [0, 1]. 5 Figure 30: the path of integration C.
1 Check if the following limits exist: z , z→0 z z 4 − 2z 2 + 1 . z→−1 z+1 (b) (a) lim lim (a) Put z = x + iy = 0. Then x − iy z . = x + iy z If in particular we choose z real, z = x, we get z = 1, z x = 1. x→0 x hence lim If we instead choose z imaginary, z = iy, then −iy z = −1, = iy z hvoraf lim y→0 iy = −1. iy Since we do not get the same limit value by the two different limits towards the same point, the limit value does not exist for z → 0, by the definition. (b) Since z 4 − 2z 2 + 1 = z 2 − 1 2 = (z + 1)2 (z − 1)2 , it follows for z = −1 that z 2 − 2z 2 + 1 = (z + 1)(z − 1)2 → 0 · (−1)2 = 0 z+1 for z → −1.
63 Complex Funktions Examples c-2 Line integrals (a) If B = 0, then eiz I(A, 0) = C 1 z+ z A dz = 1 eix 1 x+ x A dx = 0 cos x dx + i 1 x+ x A 1 sin x dx. 1 x+ x It follows that the limit A → +∞ gives (conditionally) convergent integrals and that +∞ I(∞, 0) = 0 cos x dx + i 1 x+ x +∞ 0 sin x dx = I1 + i I2 . 5 Figure 43: The curve C1 is composed of the quarter circle of radius A and centrum (1, 0) and the corresponding line segments to centrum. (b) Let C1 be the closed curve on the figure. Since the integrand is analytic inside and on C 1 , we have eiz C1 z+ 1 z dz = 0.
Complex Functions Examples c-2 – Analytic Functions by Leif Mejlbro
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