An elementary Treatise on Plane and Solid Geometry by Benjamin Peirce

By Benjamin Peirce

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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 .

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