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.

**Read or Download Complex Functions Examples c-3 - Elementary Analytic Functions and Harmonic Functions PDF**

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**Extra info for Complex Functions Examples c-3 - Elementary Analytic Functions and Harmonic Functions**

**Example text**

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}) .

Besides being a manager in the Manufacturing IT department, Kim performs triathlon at a professional level. ‘NNE Pharmaplan offers me freedom with responsibility as well as the opportunity to plan my own time. com NNE Pharmaplan is the world’s leading engineering and consultancy company focused exclusively on the pharma and biotech industries. NNE Pharmaplan is a company in the Novo Group. 4 Compute |z|=R Log z dz, z2 by considering the line integral along the curve C with the parametric description z(t) = R eit , t ∈ ] − π, π], R > 0.

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.