# Cauchy-Riemann Equations test problem

• MHB
• jaychay
In summary, Klaas stated that $u = \mathcal{Re}\left( f \right) = \mathrm{e}^x \left[ y + \cos{ \left( y \right) } \right]$ and $v = \mathcal{Im}\left( f \right) = \mathrm{e}^x \sin{ \left( y \right) }$.
jaychay

I cannot figure it out which choice is correct

What is the real part of $f(x+iy)$? That is $u(x,y)$.
What is its partial derivative with respect to $x$? That is $u_x$.

Klaas van Aarsen said:
What is the real part of $f(x+iy)$? That is $u(x,y)$.
What is its partial derivative with respect to $x$? That is $u_x$.
Is the correct answer is choice D ?
I already done it by using partial derivatives and compare them

jaychay said:
Is the correct answer is choice D ?
I already done it by using partial derivatives and compare them
It seems you are doing some online exam.
I'd rather not validate answers for something like that.

Klaas van Aarsen said:
It seems you are doing some online exam.
I'd rather not validate answers for something like that.
Sir, it's not an online exam
It's my assignment that I try to do on my own
I translate it from my math textbook in my country into English
I really want to become better at math.

If you still don't believe or you think I try to lie to you
Here is the original form of math textbook in my country

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Have you at least established what your u and v are?

As Klaas stated, \displaystyle \begin{align*} u = \mathcal{Re}\left( f \right) = \mathrm{e}^x \left[ y + \cos{ \left( y \right) } \right] \end{align*} and \displaystyle \begin{align*} v = \mathcal{Im}\left( f \right) = \mathrm{e}^x \sin{ \left( y \right) } \end{align*}. Surely you can evaluate the necessary partial derivatives and make a judgement...

Prove It said:
Have you at least established what your u and v are?

As Klaas stated, \displaystyle \begin{align*} u = \mathcal{Re}\left( f \right) = \mathrm{e}^x \left[ y + \cos{ \left( y \right) } \right] \end{align*} and \displaystyle \begin{align*} v = \mathcal{Im}\left( f \right) = \mathrm{e}^x \sin{ \left( y \right) } \end{align*}. Surely you can evaluate the necessary partial derivatives and make a judgement...
Thank you very much for guiding it for me.

## 1. What are Cauchy-Riemann equations?

Cauchy-Riemann equations are a set of conditions that describe the behavior of complex functions. They are named after mathematicians Augustin-Louis Cauchy and Bernhard Riemann, and are used to determine if a function is holomorphic (analytic) at a given point.

## 2. What is the purpose of a Cauchy-Riemann equations test problem?

The purpose of a Cauchy-Riemann equations test problem is to determine if a given function satisfies the Cauchy-Riemann conditions, and therefore is a holomorphic function. This is important in complex analysis and has applications in many areas of mathematics and physics.

## 3. What are the Cauchy-Riemann conditions?

The Cauchy-Riemann conditions state that a function is holomorphic if and only if its real and imaginary parts satisfy a set of partial differential equations. These equations involve the first order partial derivatives of the function with respect to its real and imaginary variables.

## 4. How are Cauchy-Riemann equations used in practice?

Cauchy-Riemann equations are used in practice to solve problems involving complex variables, such as in engineering, physics, and economics. They are also used in the study of conformal mappings and in the development of mathematical models.

## 5. Are there any alternative methods to test for holomorphic functions?

Yes, there are alternative methods to test for holomorphic functions, such as the Cauchy integral theorem and the Cauchy integral formula. However, the Cauchy-Riemann equations provide a straightforward and efficient way to determine if a function is holomorphic.

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