# Cauchy-Riemann conditions-Multivariable Taylor series

1. Jun 14, 2017

### Batuhan Unal

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2. Jun 14, 2017

### FactChecker

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

3. Jun 15, 2017

### Batuhan Unal

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.

Last edited: Jun 15, 2017
4. Jun 15, 2017

### FactChecker

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

5. Jun 15, 2017

### Dick

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

6. Jun 15, 2017

### Batuhan Unal

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.

7. Jun 15, 2017

### Batuhan Unal

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.

8. Jun 15, 2017

### Batuhan Unal

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.

9. Jun 15, 2017

### Dick

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?

10. Jun 15, 2017

### Batuhan Unal

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