Complex Analysis: Special Power Series

In summary, the task is to find a power series with a radius of convergence of 1 that uniformly converges for |z| ≤ 1, but its derived series does not converge at |z| = 1. Some attempts have been made using the power series for the sine function, but there is a desire to find a solution without guessing and checking multiple times.
  • #1
nateHI
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4

Homework Statement


Give an example of a power series with [itex]R=1[\itex] that converges uniformly for [itex]|z|\le 1[\itex], but such that its derived series converges nowhere for [itex]|z=1|[\itex].

Homework Equations


R is the radius of convergence and the derived series is the term by term derivative.

The Attempt at a Solution


I've tried a bunch of stuff but at the moment I'm leaning towards some variation of the power series for the sin function.
It would be nice if there was a way to solve this without guessing and checking a bunch of times.
 
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  • #2
It's OK to delete this. Posted it on accident.
 
Last edited:
  • #3
Sorry, I'm not sure why my latex commands aren't taking.
 
  • #4
You want a forward slash to terminate, not a backslash.
 

1. What is complex analysis?

Complex analysis is a branch of mathematics that deals with the study of functions of complex variables. It involves the study of complex numbers, their properties, and the behavior of functions that are defined on the complex plane.

2. What are special power series in complex analysis?

Special power series in complex analysis are infinite series that are expressed in terms of powers of a complex variable. They are used to represent complex functions and can be used to approximate solutions to problems in physics, engineering, and other fields.

3. What are some examples of special power series?

Some examples of special power series in complex analysis include the Taylor series, Laurent series, and Dirichlet series. These series have specific forms and are used to represent different types of functions.

4. How are special power series used in complex analysis?

Special power series are used in complex analysis to study the properties and behavior of complex functions. They can be used to find solutions to differential equations, calculate integrals, and determine the convergence of series.

5. What are the applications of complex analysis in real life?

Complex analysis has many applications in real life, including in physics, engineering, economics, and computer science. It is used to solve problems related to electric circuits, fluid dynamics, signal processing, and image processing, among others. It also has applications in finance, where it is used to model and analyze complex financial systems.

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