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Boost your knowledge of Complex Analysis with expert-guided homework solutions.
Complex Analysis Homework HelpQ1: Find all the roots of the complex number z = 1 + i.
See AnswerThe modulus of z = 1 + i is √2 and the argument is π/4. The roots are z₁ = √2^(1/2) * e^(iπ/8), z₂ = √2^(1/2) * e^(i(π/8 + 2πk/n)), for k = 0, 1.
Q2: Evaluate the integral ∮_C (1/(z-1)) dz, where C is the circle |z| = 2.
See AnswerUsing the residue theorem, the integral evaluates to 2πi, since the residue at z = 1 is 1.
Q3: Determine if the function f(z) = |z|² is analytic.
See AnswerThe function f(z) = |z|² is not analytic because it does not satisfy the Cauchy-Riemann equations.
Q4: Find the Taylor series expansion of f(z) = e^z about z = 0.
See AnswerThe Taylor series expansion of f(z) = e^z about z = 0 is ∑ (zⁿ/n!), where the sum runs from n = 0 to infinity.
Q5: Use the residue theorem to evaluate the integral ∮_C (sin(z)/z) dz, where C is the circle |z| = 1.
See AnswerThe residue at z = 0 is sin(0)/0 = 0. Therefore, the integral evaluates to 0.
Q6: Find the Laurent series expansion of f(z) = 1/(z - 1) around z = 0.
See AnswerThe Laurent series expansion is given by ∑ (-1)ⁿzⁿ for |z| < 1.
Q7: Show that the function f(z) = z² is conformal at z ≠ 0.
See AnswerThe function f(z) = z² is conformal at points where z ≠ 0 because it preserves angles and has a non-zero derivative.
Q8: Prove that every entire function that is bounded must be constant (Liouville's Theorem).
See AnswerBy Liouville's Theorem, since the entire function is bounded, it must be constant, as the only bounded entire functions are constants.
Q9: Determine the radius of convergence of the series ∑ (zⁿ/n!).
See AnswerThe radius of convergence is infinite, as the ratio test gives a limit of 0.
Q10: Evaluate the real part of the function f(z) = log(z) for z = 2 + 2i.
See AnswerFor z = 2 + 2i, the modulus is √8 and the argument is π/4. The real part of log(z) is log(√8) = (1/2) log(8).
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