# Complex Functions Examples c-3 - Elementary Analytic by Mejlbro L.

By Mejlbro L.

This is often the 3rd loose textbook containing examples from the speculation of complicated features. a number of the themes are examples of effortless analytic capabilities, like polynomials, fractional features, exponential capabilities and the trigonometric and the hyperbolic capabilities.

Best geometry and topology books

Famous problems of elementary geometry: the duplication of the cube, the trisection of an angle, the quadrature of the circle: an authorized translation of F. Klein's Vorträge

Greatly considered as a vintage of contemporary arithmetic, this increased model of Felix Klein's celebrated 1894 lectures makes use of modern thoughts to check 3 recognized difficulties of antiquity: doubling the amount of a dice, trisecting an perspective, and squaring a circle. contemporary scholars will locate this quantity of specific curiosity in its solutions to such questions as: below what conditions is a geometrical development attainable?

Additional resources for Complex Functions Examples c-3 - Elementary Analytic Functions and Harmonic Functions

Example text

1) Find the domains of analyticity A of f , and B of g, and sketch B. Find the derivative g (z) for z ∈ B. 2) Denote by Γ any oriented closed curve in B, and ﬁnd the value of the line integral g (z) dz. Γ Let γ denote any oriented curve in B of initial point z = −i and end point z = i. Prove that √ π 4 g (z) dz = i 2 2 sin . 8 γ 1) Clearly, A = C \ {z ∈ C | Re(z) ≤ 0, Im(z) = 0}. The exceptional set of g is given by 1 − z 3 ∈ R− ∪ {0}, hence z 3 ∈ [1, +∞[, and thus B =C\ z ∈C z = r · eiθ , r ≥ 1, θ ∈ − 2π 2π , 0, 3 3 .

2iz i w2 + 1 cos z i e +1 Since we require that tan z is deﬁned, we must have w 2 = −1. Hence, w ∈ C \ {−i , 0 , i} = Ω. com 55 Complex Functions Examples c-3 Trigonometric and hyperbolic functions Then we put w = eiz ∈ Ω into the given equation, and obtain after a rearrangement, w2 − 1 1 w2 − 1 = i w + 1 + i w2 + 1 w2 + 1 2 2 (w + 1) w w +1+w−1 = i(w + 1) · , =i w2 + 1 wr + 1 0 = i 1 + eiz − tan z = i(1 + w) − = i(w + 1) 1 + w−1 w2 + 1 where we shall solve the equation for w ∈ Ω = C \ {−i , 0 , i}.

A) It follows from |F (z)| = exp z 2 = exp x2 − y 2 + 2i xy = exp x2 − y 2 = R > 0, that x2 − y 2 = ln R ∈ R, which is the equation of a system of hyperbolas, supplied with the straight lines y = x and y = −x, both corresponding to R = 1. com 36 Complex Functions Examples c-3 The exponential function and the logarithm function 2 1 –2 –1 0 1 2 –1 –2 2 Figure 6: Some level curves F (z)| = ex −y 2 = R > 0. (b) By using polar coordinates we get the description F r eiθ = exp r2 cos 2θ · exp i r2 sin 2θ .