Exercise · ZW-HOLOMORPHIC-FUNCTION-006 · ★★☆ Intermediate
If \(f\) and \(g\) are holomorphic on \(U\), which of these functions is not necessarily holomorphic on all of \(U\)?
Choose one option.
Show answer
- D. \(f/g\)
Explanation
Sum, difference and product preserve holomorphy; the quotient fails at the points where \(g\) vanishes.
Review the concept: Holomorphic function
A complex function \(f\) is holomorphic on an open set \(U\) of the plane if it is complex differentiable at every point of \(U\).
More practice problems: holomorphic function
- ZW-HOLOMORPHIC-FUNCTION-001 Which of these functions is holomorphic on all of ℂ?
- ZW-HOLOMORPHIC-FUNCTION-002 On which set is f(z)=1/z holomorphic?
- ZW-HOLOMORPHIC-FUNCTION-003 Saying that f is holomorphic on an open set U means that:
- ZW-HOLOMORPHIC-FUNCTION-004 The function f(z)=|z|^2 is differentiable at z=0. Is it holomorphic on an open set containing 0?
- ZW-HOLOMORPHIC-FUNCTION-005 Compute f'(i) for the holomorphic function f(z)=z^2+3z.
- ZW-HOLOMORPHIC-FUNCTION-007 Let f=u+iv with u=x^2-y^2 and v=axy. For which value of a is f holomorphic?
- ZW-HOLOMORPHIC-FUNCTION-008 Compute f'(1) for f(z)=(z^2+1)^3.
- ZW-HOLOMORPHIC-FUNCTION-009 If f is holomorphic on a connected open set U and u=Ref is constant, what follows?