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

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In summary, the conversation discusses the relationship between the derivative of the imaginary part of a complex function and the derivative of the function itself. The conversation also mentions the concept of analyticity and the differentiability of a function as a real-valued function of two real variables. The question of whether a function is complex differentiable is also raised.
  • #1
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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  • #2
You can compute what it should be from the definitions, can't you? Are you sure I am f(z) is even complex differentiable?
 
  • #3
All I know is that f(z) is analytic.
 
  • #4
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?
 
  • #5
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)
 

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

What does the notation D/dz mean?

The notation D/dz represents the derivative with respect to the complex variable z. It is used to denote the rate of change of a function with respect to z.

What is Im(f(z))?

Im(f(z)) represents the imaginary part of the complex-valued function f(z). It is also known as the imaginary component or the imaginary part of the function.

What does the equation D/dz(Im(f(z))) = Im(f'(z)) mean?

This equation means that the derivative of the imaginary part of a complex-valued function with respect to z is equal to the imaginary part of the derivative of that function with respect to z.

Can this equation be applied to any complex-valued function?

Yes, this equation can be applied to any complex-valued function that is differentiable with respect to z. However, it is important to note that the functions must satisfy the Cauchy-Riemann equations for this equation to hold.

What is the significance of this equation in complex analysis?

This equation is significant in complex analysis because it relates the imaginary part of a function to its derivative. It helps in understanding the behavior of complex-valued functions and is often used in solving problems related to harmonic functions and conformal mappings.

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