Exercise · ZW-CAUCHY-RIEMANN-EQUATIONS-007 · ★★☆ Intermediate
Let \(u=x^2-y^2\). Which \(v\), with \(v(0,0)=0\), makes \(u\) and \(v\) satisfy Cauchy-Riemann?
Choose one option.
Show answer
- D. \(2xy\)
Explanation
From \(v_y=u_x=2x\) we get \(v=2xy+g(x)\), and from \(v_x=-u_y=2y\) we get \(g'=0\); with \(v(0,0)=0\), \(v=2xy\).
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-005 For f(z)=|z|^2, with u=x^2+y^2 and v=0, at which points do the Cauchy-Riemann equations 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-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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