MHB Cauchy-Riemann Equations test problem

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SUMMARY

The discussion centers on solving a problem related to the Cauchy-Riemann equations, specifically determining the real part of the function \( f(x+iy) \) and its partial derivative with respect to \( x \). The real part is defined as \( u(x,y) = \mathcal{Re}(f) = e^x [y + \cos(y)] \), while the imaginary part is \( v(x,y) = \mathcal{Im}(f) = e^x \sin(y) \). Participants emphasize the importance of evaluating the necessary partial derivatives to arrive at the correct answer, which is confirmed to be choice D.

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  • Knowledge of partial derivatives
  • Basic proficiency in mathematical notation and terminology
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jaychay
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I cannot figure it out which choice is correct
Please help me
Please!
Thank you in advance
 
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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.
 

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