Complex function that satisfies Cauchy Riemann equations

In summary, the conversation is about finding all complex numbers where the function f(z) = z cos(z-bar) is holomorphic, and the speaker is seeking help to derive the partial differential equations for this function.
  • #1
beetlez
2
0
Hi,
I am currently teaching myself complex analysis (using Stein and Shakarchi) and wondered if someone can guide me with this:

Find all the complex numbers z∈ C such that f(z)=z cos (z ̅).

[z ̅ is z-bar, the complex conjugate).

Thanks!
 
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  • #2
Hi beetlez,

Just to be clear, are you looking to find all complex numbers $z$ at which $f$ holomorphic?
 
  • #3
Euge said:
Hi beetlez,

Just to be clear, are you looking to find all complex numbers $z$ at which $f$ holomorphic?

Hi, no actually just assuming that the function is differentiable, I just wanted help to derive the partial differential equations (du/dx, du/dy, dv/dx and dv/dy).
 

FAQ: Complex function that satisfies Cauchy Riemann equations

What are Cauchy Riemann equations?

Cauchy Riemann equations are a set of two partial differential equations that define the analyticity of a complex-valued function. They are named after the mathematicians Augustin-Louis Cauchy and Georg Friedrich Bernhard Riemann.

What does it mean for a function to satisfy Cauchy Riemann equations?

If a function satisfies Cauchy Riemann equations, it means that the function is differentiable at a given point and has a complex derivative at that point.

What is the significance of functions that satisfy Cauchy Riemann equations?

Functions that satisfy Cauchy Riemann equations are important in complex analysis because they are considered to be the most well-behaved functions in the complex plane. They have many useful properties and are crucial in the study of complex functions.

How can one determine if a function satisfies Cauchy Riemann equations?

To determine if a function satisfies Cauchy Riemann equations, one needs to check if the function satisfies both the real and imaginary parts of the equations. This involves taking partial derivatives of the function and equating them to zero.

What are some applications of Cauchy Riemann equations?

Cauchy Riemann equations have various applications in mathematics and physics, including in the study of fluid dynamics, electromagnetic theory, and harmonic functions. They are also used in solving differential equations and in the development of numerical methods for solving complex problems.

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