6. Further Pure Mathematics
Complex Analysis — Quiz
Test your understanding of complex analysis with 5 practice questions.
Practice Questions
Question 1
Which definition correctly describes complex differentiability at a point $z_0$?
Question 2
Determine whether the function $f(z)=u(x,y)+iv(x,y)$ with $u(x,y)=x^2-y^2$ and $v(x,y)=x^2+y^2$ is analytic. Use the Cauchy–Riemann equations to justify your answer.
Question 3
Find a harmonic conjugate $v(x,y)$ for the real function $u(x,y)=2xy$ so that $f(z)=u+iv$ is analytic. Determine $v(x,y)$ up to an additive constant.
Question 4
Evaluate the contour integral
$I=\oint_{|z-i|=1}\frac{dz}{z^2+1}$
taken counterclockwise around the circle centered at $z=i$ with radius 1.
$I=\oint_{|z-i|=1}\frac{dz}{z^2+1}$
taken counterclockwise around the circle centered at $z=i$ with radius 1.
Question 5
What is the radius of convergence of the power series
$\sum_{n=0}^\infty n!\,z^n$
centered at $z=0$?
$\sum_{n=0}^\infty n!\,z^n$
centered at $z=0$?