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Huns

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Is this function analytic or not? Please explain

Is this function analytic or not? Please explain

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- Thread starter Huns
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In summary, an analytic function is a complex-valued function that is differentiable at every point within its domain. It can be represented by a power series and is a combination of a real part and an imaginary part. However, it is only analytic if certain conditions are met. Analytic functions are important in science as they allow for the modeling and analysis of complex systems, and their properties make them useful for various applications. To determine if a function is analytic, one can check if it satisfies the Cauchy-Riemann equations.

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Huns

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Is this function analytic or not? Please explain

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Svein

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Well, it's the definition of log(z).

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Huns

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so how to split ln (z) and arg (z) for determining analyticity?Svein said:Well, it's the definition of log(z).

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Svein

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log(z) is analytic in any simply connected domain that does not include 0.Huns said:so how to split ln (z) and arg (z) for determining analyticity?

By the way, the expression ln|z| is the

An analytic function is a complex-valued function that is differentiable at every point within its domain. It can be represented by a power series of the form f(z) = ∑c_{n}(z-a)^{n}, where c_{n} are complex coefficients and a is a constant.

f(z)+i arg(z) is a complex-valued function that combines the real part of the function f(z) with the imaginary part of the argument of z. It can be written as f(z) = u(x,y) + iv(x,y), where u(x,y) and v(x,y) are the real and imaginary parts, respectively.

No, f(z)+i arg(z) is not always an analytic function. It is only analytic if u(x,y) and v(x,y) satisfy the Cauchy-Riemann equations, which are a set of necessary and sufficient conditions for a complex-valued function to be differentiable.

Analytic functions are important in science because they provide a way to model and analyze complex systems, such as in physics, engineering, and economics. They also have many useful properties, such as being infinitely differentiable and having a unique analytic continuation.

To determine if f(z)+i arg(z) is an analytic function, one can check if the Cauchy-Riemann equations are satisfied. If they are, then the function is analytic and can be represented by a power series. If they are not satisfied, then the function is not analytic and cannot be represented by a power series.

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