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Cauchy-Riemann conditions-Multivariable Taylor series

  1. Jun 14, 2017 #1
    İ couldn't understand the last operation, please help me.
     

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  3. Jun 14, 2017 #2

    FactChecker

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    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.
     
  4. Jun 15, 2017 #3
    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.
     
    Last edited: Jun 15, 2017
  5. Jun 15, 2017 #4

    FactChecker

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    Oh. Sorry. I answered the easy part and totally overlooked the hard part. I don't see an answer to that part now.
     
  6. Jun 15, 2017 #5

    Dick

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    Work out the binomial expansion of ##[(x+iy)-(x_0+iy_0)]^n##. Separate into real and imaginary parts inside the brackets first.
     
  7. Jun 15, 2017 #6
    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 #7
    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 #8
    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.
     
  10. Jun 15, 2017 #9

    Dick

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    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?
     
  11. Jun 15, 2017 #10
    Sorry, my head has gone to the infinity binomial series. My problem has been solved, thanks you a lot.
     
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