Showing Cauchy-Riemann Equations Hold & Derivative of Function

[andrey21]

I have been asked to show that the Cauchy-Riemann equations hold for a function

I have managed to do this successfully and now have to show the derivative of this function.



Using the following formula

f'(z) = du/dx + i dv/dx

where du/dx = 2x +1 and dv/dx = 2y

so this gives me:

2x+1 + 2iy

which can be written as:

2z+1

correct?

 

[Char. Limit]

I have been asked to show that the Cauchy-Riemann equations hold for a function

I have managed to do this successfully and now have to show the derivative of this function.



Using the following formula

f'(z) = du/dx + i dv/dx

where du/dx = 2x +1 and dv/dx = 2y

so this gives me:

2x+1 + 2iy

which can be written as:

2z+1

correct?

As long as z=x+iy, there's no problem with that. At least I think so.

 

[andrey21]

Yes that's the way I've been taught to use z=x+iy. Thank you Char.limit

 

[lanedance]

except its worth noting f'(z) does not just mean df/dx, it is a more stringent condition, however if the function is holomorphic, they will be equal.

df/dx repsents the derivative along a path parallel to the real axis