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

This can be the 3rd loose textbook containing examples from the speculation of complicated capabilities. the various themes are examples of hassle-free analytic capabilities, like polynomials, fractional services, exponential services and the trigonometric and the hyperbolic features.

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2+ 3 Hence z = = √ 1 log i 3±2 i ⎧ 1 √ π ⎪ ⎪ ⎨ i ln(2+ 3)+i 2 + 2pπ = ⎪ ⎪ ⎩ 1 − ln(2+ √3)+i − π + 2pπ i 2 √ π + 2pπ−i ln(2+ 3), 2 = − √ π + 2pπ + i ln(2+ 3 =), 2 and summing up, z = 2pπ ± √ π − i ln(2 + 3) , 2 p ∈ Z. 17 Given the functions f (z) = 1 −1 z2 and g(z) = L0 (z) where L0 denotes the branch of the logarithm, which is deﬁned by L0 = ln |z| + i arg0 (z) where arg0 (z) ∈ ]0, 2π]. Find the domains of analyticity of the functions f , g and h = g ◦ f . Clearly, f is analytic in C \ {0}, and g is analytic in C \ {z ∈ C | Im(z) = 0 and Re(z) ≥ 0} = C \ (R+ ∪ {0}) .

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This is only an easy exercise in ﬁnding a primitive, because the integrand of course is analytic in C, and hence independent of the path of integration: 3i cos2 (π i z) sin2 (π i z) dz 1 4 = 0 3i sin2 (2π i z) dz = 0 1 4 3i 0 1 {1 − cos(4π i z)} dz 2 1 3i 1 1 1 {1 − cosh(4π z)} dz = · 3i − · [sinh(4π z]3i 0 8 0 8 8 4π 3i 1 3i i 3i − sinh(12πi) = − sin(12π) = . 6 Describe the streamlines of the complex potential F (z) = sinh z, y ∈ [0, π]. The stream function is here given by ψ(x, y) = Im(F (z)) = Im(sinh z) = cosh x · sin y.