How do you show that a complex function is analytical?

Thread starter twotaileddemon
Start date Mar 17, 2010
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Analytical Complex Complex function Function

In summary, a function is considered analytic if it is differentiable at every point in its domain, meaning its derivative exists and is continuous. The Cauchy-Riemann equation is a set of conditions used to show analyticity, stating that the partial derivatives of the function must satisfy a specific relationship. A function must also be holomorphic to be analytic. The Taylor series expansion and other methods such as the Cauchy integral formula and power series can also be used to prove analyticity.

Mar 17, 2010

twotaileddemon

Like if I wanted to show how sin z, cos z, or e^z are analytical, what is the general process I have to do? Can I use the cauchy - riemann relations somehow?

(where z = x + iy is complex)

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Mar 17, 2010

LCKurtz

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Yes. For example, e^{z =}e^x+iy = e^xe^iy =e^x(cos(y) + i sin(y)) so

u = e^xcos(y) and v = e^xsin(y)

and you can Cauchy-Riemann away.

Mar 18, 2010

twotaileddemon

Thanks! :)

Related to How do you show that a complex function is analytical?

1. How do you determine if a function is analytic?

An analytic function is one that is differentiable at every point in its domain. This means that the function's derivative exists and is continuous at every point. A function that is not differentiable at a point is not considered analytic.

2. What is the Cauchy-Riemann equation and how is it used to show analyticity?

The Cauchy-Riemann equation is a set of necessary and sufficient conditions for a function to be analytic. It states that the partial derivatives of the function with respect to the real and imaginary components of its input must satisfy a specific relationship. If the Cauchy-Riemann equation holds, then the function is analytic.

3. Can a function be analytic but not holomorphic?

No, a function that is analytic must also be holomorphic. This means that the function is not only differentiable at every point in its domain, but its derivative is also continuous at every point. Holomorphic functions are important in complex analysis and have many useful properties.

4. What is the significance of the Taylor series in determining analyticity?

The Taylor series expansion of a function is used to show that the function is analytic. If a function can be expressed as a Taylor series, it means that the function is infinitely differentiable at every point in its domain. This is a necessary condition for analyticity.

5. Are there any other methods for proving analyticity besides the Cauchy-Riemann equation and Taylor series?

Yes, there are several other methods for proving analyticity, such as using the Cauchy integral formula or the power series expansion of a function. Additionally, a function can be shown to be analytic if it satisfies the Laplace equation or if it has a Laurent series representation. These methods may be more applicable in certain situations and can be used as alternatives to the Cauchy-Riemann equation and Taylor series.