Verify the Cauchy-Riemann equations for f(z)=z².
Answer and explanation
u=x²−y² and v=2xy. Then u_x=2x=v_y and u_y=−2y=−v_x.
Credit both equations and smoothness of the partials.
Subjects
Complex analysis
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Sample exercises
For f(z)=z³, find the real derivative value f′(2).
Answer and explanation
12
f′(z)=3z², so f′(2)=12.
Is complex conjugation f(z)=conjugate(z) holomorphic on an open subset of C?
- It is a polynomial in z
- No
- Only because it is continuous
- Yes everywhere
Answer and explanation
No
For u=x,v=−y, u_x=1 but v_y=−1, so the Cauchy-Riemann equations fail.
Use directional difference quotients to show conjugation is not complex differentiable at zero.
Answer and explanation
Along real h, conjugate(h)/h=1; along imaginary h, the quotient is −1. The limits disagree.
Credit two directions with distinct limiting quotients.
What does a power series define inside its radius of convergence?
- A function with no derivative
- A necessarily constant function
- Only a function on the real axis
- A holomorphic function
Answer and explanation
A holomorphic function
Complex power series can be differentiated termwise inside their convergence disk.
For f(z)=z² at z=i, write f′(i)=a i and find a.
Answer and explanation
2
f′(z)=2z, hence 2i.
Study notes
Holomorphic functions and Cauchy-Riemann equations
A derivative must have the same limit along every complex direction.
Advanced universityHolomorphic functions and Cauchy-Riemann equations · revision notes
A derivative must have the same limit along every complex direction.
Advanced universityPoles, residues and contour integrals
The coefficient of (z−a)^(−1) in the Laurent expansion is the residue.
Advanced universityPoles, residues and contour integrals · revision notes
The coefficient of (z−a)^(−1) in the Laurent expansion is the residue.
Advanced universityResidue at a simple pole
The numerator is analytic near a and the pole is simple.
Advanced universityResidue at a simple pole · formula reference
The numerator is analytic near a and the pole is simple.
Advanced universityResidue at a second-order pole
Differentiate the analytic numerator once; its value p(a) is not the residue for a double pole.
Advanced universityResidue at a second-order pole · formula reference
Differentiate the analytic numerator once; its value p(a) is not the residue for a double pole.
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