Complex analysis taylor series Q

In summary, a Taylor series in complex analysis is an infinite sum of terms that approximates the behavior of a complex-valued function around a specific point. The formula for the Taylor series involves the function, its derivatives, and the point of approximation. This series is used to approximate functions in various fields, such as physics and engineering. A complex power series is a type of Taylor series where the variable is raised to a complex power. The radius of convergence for a Taylor series in complex analysis is the distance from the point of approximation to the nearest singularity.
  • #1
ElDavidas
80
0
hi, I'm wondering if someone can help me out with this question:

"What are the first two non-zero terms of the Taylor series

[tex]f(z) = \frac {sin(z)} {1 - z^4} [/tex] expanded about z = 0.

(Don't use any differentiation. Just cross multiply and do mental arithmetic)"

I know the formula for a Taylor series but I can't see how to do this with just cross multiplying and mental arithmetic.

Thanks
 
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  • #2
This is a product of two functions: sin(z) and 1/(1-z^4).
If you know the Taylor series of both, you can find their product by 'expanding the brackets'
 

Related to Complex analysis taylor series Q

1. What is a Taylor series in complex analysis?

A Taylor series is a representation of a function as an infinite sum of terms, where each term is a polynomial multiplied by a power of the variable. In complex analysis, the Taylor series is used to approximate the behavior of a complex-valued function around a specific point.

2. What is the formula for the Taylor series in complex analysis?

The formula for the Taylor series in complex analysis is given by f(z) = f(a) + f'(a)(z-a) + \frac{f''(a)}{2!}(z-a)^2 + \frac{f'''(a)}{3!}(z-a)^3 + ..., where f(z) is the complex-valued function, a is the point around which the approximation is made, and f'(a), f''(a), f'''(a), ... are the derivatives of f(z) evaluated at a.

3. How is the Taylor series used in complex analysis?

The Taylor series is used in complex analysis to approximate the behavior of a function near a particular point. By taking more and more terms in the series, the approximation becomes more accurate. This is useful in many applications, such as in physics and engineering, where complex functions are often encountered.

4. What is a complex power series?

A complex power series is a type of Taylor series where the variable is raised to a complex power. In other words, the terms in the series are of the form a_n(z-a)^n, where a_n is a complex number and n is the power of the variable. These series can be used to represent more general complex-valued functions than traditional Taylor series.

5. What is the radius of convergence for a Taylor series in complex analysis?

The radius of convergence for a Taylor series in complex analysis is the distance from the point around which the approximation is made to the nearest singularity (point where the function is not defined). If the distance is infinite, the series converges everywhere, and if the distance is zero, the series converges only at the point a. Otherwise, the series converges within a certain disk around the point a.

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