Cauchy-Riemann conditions-Multivariable Taylor series

[Batuhan Unal]

İ couldn't understand the last operation, please help me.

 

[FactChecker]

Instead of taking the k partial derivatives wrt y, they replace them with i^k times k more partial derivatives wrt x as shown in the equation right above.

 

[Batuhan Unal]

Thank you for the answer.
I understood that but then he somehow gets rid of the second ∑ that which the sums with the k and n terms. Actually i i have congested at there.

 

[FactChecker]

Oh. Sorry. I answered the easy part and totally overlooked the hard part. I don't see an answer to that part now.

 

[Dick]

Thank you for the answer.
I understood that but then he somehow gets rid of the second ∑ that which the sums with the k and n terms. Actually i i have congested at there.

Work out the binomial expansion of $[(x+iy)-(x_0+iy_0)]^n$. Separate into real and imaginary parts inside the brackets first.

 

[Batuhan Unal]

Work out the binomial expansion of $[(x+iy)-(x_0+iy_0)]^n$. Separate into real and imaginary parts inside the brackets first.

Thank you, now i can see from the n!/(k!(n-k)! terms that how the second sigma notatian has gone. But there i need to do a multivariable binomial series expansion but i can't do it.

 

[Batuhan Unal]

Work out the binomial expansion of $[(x+iy)-(x_0+iy_0)]^n$. Separate into real and imaginary parts inside the brackets first.

Thank you, now i can see from the n!/(k!(n-k)! terms that how the second sigma notatian has gone. But there i need to do a multivariable binomial series expansion but i can't do it.

 

[Batuhan Unal]

Work out the binomial expansion of $[(x+iy)-(x_0+iy_0)]^n$. Separate into real and imaginary parts inside the brackets first.

Thank you, now i can see from the n!/(k!(n-k)! terms that how the second sigma notatian has gone. But there i need to do a multivariable binomial series expansion but i can't do it.

 

[Dick]

Thank you, now i can see from the n!/(k!(n-k)! terms that how the second sigma notatian has gone. But there i need to do a multivariable binomial series expansion but i can't do it.

It's not very clear what you mean. You just want to expand $(a+b)^n$ where $a=(x-x_0)$ and $b=i(y-y_0)$. It's the perfectly normal type of binomial expansion. Or are you asking about something else?

 

[Batuhan Unal]

It's not very clear what you mean. You just want to expand $(a+b)^n$ where $a=(x-x_0)$ and $b=i(y-y_0)$. It's the perfectly normal type of binomial expansion. Or are you asking about something else?

Sorry, my head has gone to the infinity binomial series. My problem has been solved, thanks you a lot.