Analysis: Absolute convergence, rearranging terms, power series question

In summary, the conversation discusses proving the analyticity of the product of two analytic functions. The forum poster shares their attempt at a solution, which involves using the fact that both functions must be convergent in their respective domains. The last two lines of the professor's statement may be referring to rearranging the terms of the power series representation of the product to show that it is also in the form of a power series, thus proving its analyticity.
  • #1
pcvt
18
0

Homework Statement


I can't figure out what my professor means in the last two lines:

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I'm trying to prove why the product of two analytic functions is analytic, and I think that I am going to need to use a similar construction.

The Attempt at a Solution



So far to prove the product is analytic, I have said that for
f(x) convergent for x<R1
g(x) convergent for x<R2
Then, (f*g)(x) is convergent for x<min(R1,R2), then, I think I can say that for some rearrnagment of terms, (f*g)(x) = sum n=0 to infinity of (c_n x^n),

but I'm not sure if or why I could make that last step.

Thanks!
 
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  • #2


It seems like you are on the right track with your proof. The last two lines of your professor's statement may be referring to the fact that for the product of two analytic functions to be analytic, both functions must be convergent in their respective domains. This means that the power series representation of the product must also be convergent. Therefore, by rearranging the terms of the power series, you can show that the product is also in the form of a power series, which is a characteristic of analytic functions.

I hope this helps clarify the last two lines for you. Keep up the good work!
 

FAQ: Analysis: Absolute convergence, rearranging terms, power series question

What is absolute convergence?

Absolute convergence is a property of a series of numbers or terms where the sum of the absolute values of the terms converges to a finite number. In other words, the series does not depend on the order in which the terms are added and will always converge to the same value.

How is absolute convergence different from conditional convergence?

Absolute convergence and conditional convergence are two different types of convergence for series. Absolute convergence means that the series converges regardless of the order of the terms, while conditional convergence means that the series only converges when certain conditions are met, such as the terms being added in a specific order.

What is the importance of rearranging terms in a series?

Rearranging terms in a series can change the value of the series, especially when the series is conditionally convergent. This is because rearranging the terms can change the order in which they are added, which can affect the overall sum. Therefore, it is important to understand the properties of a series and whether it is absolutely or conditionally convergent before rearranging terms.

How can power series be used in analysis?

Power series are infinite series of the form ∑n=0∞ an(x-c)n, where an are coefficients and c is a constant. They can be used in analysis to approximate functions, solve differential equations, and study the properties of functions, such as continuity and differentiability.

Are there any real-world applications of absolute convergence and rearranging terms?

Absolute convergence and rearranging terms have real-world applications in fields such as engineering, physics, and economics. For example, in engineering, understanding the properties of series can help in designing stable structures. In physics, power series are used to approximate physical phenomena, such as the motion of a pendulum. In economics, absolute convergence and rearranging terms are used in financial calculations and risk management.

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