Exercise · ZW-CAUCHY-RIEMANN-EQUATIONS-005 · ★★☆ Intermediate
For \(f(z)=|z|^2\), with \(u=x^2+y^2\) and \(v=0\), at which points do the Cauchy-Riemann equations hold?
Choose one option.
Show answer
- C. Only at \(z=0\)
Explanation
The equations require \(2x=0\) and \(2y=0\), which happens only at the origin.
Review the concept: Cauchy-Riemann equations
The two equalities between the partial derivatives of the real part \(u\) and the imaginary part \(v\) of a function \(f=u+iv\) that is differentiable at a point.
\[u_x=v_y,\qquad u_y=-v_x\]
More practice problems: cauchy-riemann equations
- ZW-CAUCHY-RIEMANN-EQUATIONS-001 For f(z)=z^2 we have u=x^2-y^2 and v=2xy. What is u_x?
- ZW-CAUCHY-RIEMANN-EQUATIONS-002 Which is the correct pair of Cauchy-Riemann equations?
- ZW-CAUCHY-RIEMANN-EQUATIONS-003 For f(z)=e^z, with u=e^xcos y and v=e^xsin y, what is v_y?
- ZW-CAUCHY-RIEMANN-EQUATIONS-004 For f(z)=bar z, with u=x and v=-y, at which points does u_x=v_y hold?
- ZW-CAUCHY-RIEMANN-EQUATIONS-006 If f=u+iv is differentiable at z, then f'(z)=u_x+iv_x. Compute f'(1+i) for f(z)=z^2.
- ZW-CAUCHY-RIEMANN-EQUATIONS-007 Let u=x^2-y^2. Which v, with v(0,0)=0, makes u and v satisfy Cauchy-Riemann?
- ZW-CAUCHY-RIEMANN-EQUATIONS-008 Let u=e^xcos y. Which v satisfies Cauchy-Riemann together with u?
- ZW-CAUCHY-RIEMANN-EQUATIONS-009 If f=u+iv satisfies the Cauchy-Riemann equations at a point, what can be asserted?
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