Is d/dz(Im(f(z))) always equal to Im(f'(z)) for an analytic function f(z)?

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Discussion Overview

The discussion centers on the relationship between the derivative of the imaginary part of an analytic function \( f(z) \) and the imaginary part of the derivative of that function. Participants explore whether the equation \( \frac{d}{dz}(\text{Im}(f(z))) = \text{Im}(f'(z)) \) holds true, particularly under the condition that \( f(z) \) is analytic.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether \( \frac{d}{dz}(\text{Im}(f(z))) \) equals \( \text{Im}(f'(z)) \) and seeks clarification on the implications of \( \text{Im}(f'(z)) > 0 \) regarding the behavior of \( \text{Im}(f(z)) \).
  • Another participant suggests that the definitions should clarify the relationship and questions the assumption of \( f(z) \) being complex differentiable.
  • A participant asserts that \( f(z) \) is analytic, implying that both \( f \) and its real and imaginary parts are differentiable as real-valued functions of two real variables.
  • One participant proposes to express \( f(z) \) in terms of its real and imaginary components and to compute both sides of the equation to analyze the relationship.
  • Another participant points out that while \( f \) and its components are differentiable as real-valued functions, this does not necessarily imply the same for complex variables, raising a question about the treatment of differentiation in this context.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between the derivatives and the conditions under which they hold. There is no consensus on whether the equation \( \frac{d}{dz}(\text{Im}(f(z))) = \text{Im}(f'(z)) \) is universally valid for analytic functions.

Contextual Notes

Participants note the importance of definitions and the context of differentiability, particularly in relation to complex variables versus real-valued functions. The discussion highlights potential limitations in assumptions regarding the nature of \( f(z) \) and its derivatives.

bmanbs2
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So does d/dz(Im(f(z))) = Im(f'(z))?

For context, if Im(f'(z)) > 0, is Im(f(z)) always increasing and one-to-one?

f(z) is analytic
 
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You can compute what it should be from the definitions, can't you? Are you sure I am f(z) is even complex differentiable?
 
All I know is that f(z) is analytic.
 
You didn't say that before! But, given that f is analytic, f and its real and imaginary parts are certainly differentiable- now, let f(z)= g(z)+ ih(z) and simply write out the two sides of your equation. What is Im(df/dz)? What is d(Im(f(z))/dz?
 
HallsofIvy said:
But, given that f is analytic, f and its real and imaginary parts are certainly differentiable
... as real-valued functions of two real variables (the real and imaginary parts of z), but not necessarily as a function of one complex variable.


edit: or are you thinking about the differential-geometry-like treatment of differentiation where, e.g., df splits into a linear combination of [itex]dz[/itex] and [itex]d\bar{z}[/itex], and such? (this is directed both at HallsofIvy and the original poster)
 

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