# How Do You Calculate the Coefficients and Convergence of a Complex Power Series?

In summary, the power series for f'(z) is ∑(j+1).a_(j+1).z^j, which converges everywhere. The power series for f^2 is ∑a_mn.z^m, where the coefficients are given by a_mn = ∑a_k.a_(m-k). This series also converges everywhere. Using the conditions f(0) = 0 and f'(0) = 1, we can find the values of a_0, a_1, a_2, a_3, a_4, a_5 to be 0, 1, 0, -1/6, 0, and 1/120

## Homework Statement

Suppose that f(z) = ∑a_j.z^j for all complex z, the sum goes from j=0 to infinity.

(a) Find the power series expansion for f'
(b) Where does it converge?
(c) Find the power series expansion for f^2
(d) Where does it converge?
(e) Suppose that f'(x)^2 + f(x)^2 = 1, f(0) = 0, f'(0) = 1

Find a_0, a_1, a_2, a_3, a_4, a_5

## The Attempt at a Solution

Hi everyone, here's my attempt. I'm not sure if it's right though, so I would really appreciate it if you could please take a look at it:

(a) f'(z) = ∑j.a_j.z^(j-1), sum from 0 to infinity

= ∑j.a_j.z^(j-1), sum from 1 to infinity

= ∑(j+1).a_(j+1).z^j, sum from 0 to infinity

(b) We don't what the values of a_j are, so we can only say with definity that it converges for z = 0

(c) f^2 = (a_0)^2 + (a_0.a_1)z + (a_0.a_2)z^2 + ...
+ (a_1.a_0)z + (a_1.a_1)z^2 + ... etc.

(d) Again, we can only say it converges for definite where z = 0.

(e) f(0) = 0
i.e.
a_0 + a_1(0) + a_2(0) +... = 0

i.e. a_0 = 0

f'(0) = 1
i.e.
a_1 + 2.a_2.z + 3.a_3.z^2 +... = 1

The fact it is equal to 1 means it converges, but, as above, this can only happen if z = 0
i.e. a_1 = 1

Calculate f'(x)^2 = 1 + z(4.a_2) + z^2(6.a_3 + 4.a_2.a_2) + ...

and f(x)^2 = z^2(a_1.a_1) + z^3(2.a_1.a_2) + ...

The coefficients of the z terms in f'(x)^2 must be equal to minus the coefficients of the corresponding z terms in f(x)^2.

So we find that:

a_2 = 0
a_3 = -1/6
a_4 = 0
a_5 = 1/120

---
Regarding the convergence in (b) and (d), am I correct in what I say, in that there is no way of knowing what the a_j's are, so the only way we can know the series converge is when z = 0?

Also, have I multiplied the series correctly?

I'm sure I must have done something wrong, as it's a question from a past paper in my complex analysis class, but I don't seem to have used any complex analysis here...

Thanks for any help!

Hi everyone, here's my attempt. I'm not sure if it's right though, so I would really appreciate it if you could please take a look at it:

(a) f'(z) = ∑j.a_j.z^(j-1), sum from 0 to infinity

= ∑j.a_j.z^(j-1), sum from 1 to infinity

= ∑(j+1).a_(j+1).z^j, sum from 0 to infinity

(b) We don't what the values of a_j are, so we can only say with definity that it converges for z = 0
You can do better than that. Try applying the ratio test and use the fact the original series converges everywhere.
(c) f^2 = (a_0)^2 + (a_0.a_1)z + (a_0.a_2)z^2 + ...
+ (a_1.a_0)z + (a_1.a_1)z^2 + ... etc.
You should collect terms. There's a pattern to the coefficient of zn.
(d) Again, we can only say it converges for definite where z = 0.
Again, you should be able to do better than that. Do you know any theorems about convergence of products of series?

## 1. What is a complex power series?

A complex power series is a mathematical series that involves complex numbers, which are numbers that have a real part and an imaginary part. The series is typically written in the form of a polynomial with infinitely many terms, each of which is multiplied by a complex number raised to a certain power.

## 2. How is a complex power series different from a regular power series?

A regular power series only involves real numbers, while a complex power series includes both real and imaginary numbers. This allows for a wider range of possible values and solutions.

## 3. What is the purpose of using a complex power series in mathematical equations?

Complex power series are useful in approximating functions that are not easily expressed in simpler terms. They can also be used to solve differential equations and study the behavior of complex functions.

## 4. How is a complex power series evaluated?

The evaluation of a complex power series involves finding the values of the coefficients and determining the radius of convergence. This can be done through various methods, such as the ratio test or the Cauchy-Hadamard theorem.

## 5. Can a complex power series converge to more than one value?

Yes, a complex power series can converge to multiple values depending on the value of the input. This is known as multivalued convergence and is a unique property of complex power series.

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