A complex function 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖 𝑣(𝑥, 𝑦) and its complex conjugate 𝑓∗(𝑧) = 𝑢(𝑥, 𝑦) − 𝑖 𝑣(𝑥, 𝑦) are both analytic in the entire complex plane

August 29, 2023

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A complex function 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖 𝑣(𝑥, 𝑦) and its complex conjugate 𝑓∗(𝑧) = 𝑢(𝑥, 𝑦) − 𝑖 𝑣(𝑥, 𝑦) are both analytic in the entire complex plane

Q. A complex function 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖 𝑣(𝑥, 𝑦) and its complex conjugate 𝑓^∗(𝑧) = 𝑢(𝑥, 𝑦) − 𝑖 𝑣(𝑥, 𝑦) are both analytic in the entire complex plane, where 𝑧 = 𝑥 + 𝑖 𝑦 and 𝑖 = √−1 . The function 𝑓 is then given by

Sol:

f(z) = u(x, y) + i v(x, y)

f*(z) = u (x, y) – i v(x, y)

f(z) is analytic function

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A complex function 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖 𝑣(𝑥, 𝑦) and its complex conjugate 𝑓∗(𝑧) = 𝑢(𝑥, 𝑦) − 𝑖 𝑣(𝑥, 𝑦) are both analytic in the entire complex plane

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The function 𝑝(𝑥) is given by 𝑝(𝑥) = 𝐴/𝑥𝜇 where 𝐴 and 𝜇 are constants with 𝜇 > 1 and 1 ≤ 𝑥 < ∞ and 𝑝(𝑥) = 0 for −∞ < 𝑥 < 1

A complex function 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖 𝑣(𝑥, 𝑦) and its complex conjugate 𝑓∗(𝑧) = 𝑢(𝑥, 𝑦) − 𝑖 𝑣(𝑥, 𝑦) are both analytic in the entire complex plane

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The forward path transfer function L(s) of the control system shown in Figure (a) has the asymptotic Bode plot shown

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