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Batuhan Unal
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İ couldn't understand the last operation, please help me.
Batuhan Unal said: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.
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 said: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 said: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 said: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 said: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.
Sorry, my head has gone to the infinity binomial series. My problem has been solved, thanks you a lot.Dick said: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?
The Cauchy-Riemann conditions are a set of necessary and sufficient conditions for a complex-valued function to be differentiable at a point. They state that the partial derivatives of the real and imaginary parts of the function must satisfy a set of equations, known as the Cauchy-Riemann equations.
The Cauchy-Riemann conditions are important because they determine when a complex-valued function is differentiable and can be represented by a power series, which is a fundamental concept in complex analysis. They also have applications in physics, engineering, and other fields.
The Multivariable Taylor series is a method for approximating a multivariable function using a polynomial. It is an extension of the Taylor series, which is used to approximate single-variable functions. The Multivariable Taylor series takes into account the partial derivatives of the function at a given point, allowing for a more accurate approximation.
The Cauchy-Riemann conditions are necessary for a complex-valued function to be differentiable, and the Multivariable Taylor series requires the function to be differentiable in order to be approximated. Therefore, the Cauchy-Riemann conditions are a crucial component in the derivation and use of the Multivariable Taylor series.
The Cauchy-Riemann conditions and Multivariable Taylor series have applications in many fields, such as fluid mechanics, electromagnetism, and signal processing. They are used to model and analyze complex systems, as well as to solve differential equations. In addition, they are essential in the development of numerical methods for solving problems in science and engineering.