Division of complex power series

However, it's important to note that the power series for tan(z) will only converge for certain values of z, just like the power series for sin(z) and cos(z). In summary, to find the power series for tan(z), you can divide the power series for sin(z) by the power series for cos(z), but be aware of the convergence conditions for both series.
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Homework Statement


Find tan(z) up to the z^7 term, where tan(z) = sin(z)/cos(z)


Homework Equations


sin(z) = z - z^3/3! + z^5/5! - z^7/7! + ...

cos(z) = 1 - z^2/2! + z^4/4! - z^6/6! + ...


The Attempt at a Solution


Hi,
Seeing as sin and cos have the same power series as for when they are real, can you just divide the complex polynomials?

i.e. (z - z^3/3! + z^5/5! - z^7/7! + ...) / (1 - z^2/2! + z^4/4! - z^6/6! + ...) = z + z^3/3 + 2z^5/15 + 17z^7/315 + ...

which is tan(z)? (Assuming it has the same complex power series as real power series, considering sin and cos do?)

Thanks for any help
 
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  • #2
Yes, you can do it that way.
 

1. What is the definition of a complex power series?

A complex power series is an infinite sum of terms of the form an(z-c)n, where an are complex coefficients, z is a complex variable, and c is a complex constant.

2. How is the radius of convergence determined for a complex power series?

The radius of convergence for a complex power series is determined by applying the ratio test. This involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If this limit is less than 1, then the series will converge within a certain radius around the center point c.

3. Can a complex power series have multiple points of convergence?

No, a complex power series can only have one point of convergence, which is the center point c. This is because the radius of convergence is defined as the distance from c where the series converges. If there were multiple points of convergence, the radius of convergence would be different for each point.

4. What is the difference between a power series and a complex power series?

A power series is a series of terms with only real coefficients, while a complex power series includes terms with complex coefficients. Additionally, the variable z in a complex power series can take on complex values, while in a power series it is limited to real values.

5. How is the division of complex power series performed?

The division of complex power series involves using the geometric series formula to rewrite the divisor as a simpler series. Then, the quotient is found by multiplying the dividend by the inverse of the simplified divisor. This process is similar to long division for polynomials.

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