Exercise · ZW-CAUCHY-RIEMANN-EQUATIONS-001 · ★☆☆ Basic
For \(f(z)=z^2\) we have \(u=x^2-y^2\) and \(v=2xy\). What is \(u_x\)?
Choose one option.
Show answer
- C. \(2x\)
Explanation
Differentiate \(u=x^2-y^2\) with respect to \(x\), treating \(y\) as a constant: \(u_x=2x\). It matches \(v_y=2x\).
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-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-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?