My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum


Reply
 
LinkBack Thread Tools Display Modes
April 16th, 2013, 07:26 PM   #1
Newbie
 
Joined: Feb 2013

Posts: 15
Thanks: 0

Strange analytic function

If r E R, we can de fine a very strange function fr by the following process. Express r as a decimal
as r0.a1a2a3 ... where r0 is an integer and each of the ai is a digit between 0 and 9. Then the function is defined as

fr(x) = r0 + infinity/E/n = 1 (an/n!) x^n

Prove that this function is analytic on R for any choice of r. What kind of functions do we get if r is
an integer? If r is rational?
bach71 is offline  
 
April 17th, 2013, 05:22 AM   #2
Math Team
 
Joined: Sep 2007

Posts: 2,409
Thanks: 6

Re: Strange analytic function

I am afraid you will have to explain what you mean by "infinity/E/n" for this to make any sense. Do you mean "sum for n= 1 to infinity"?
HallsofIvy is offline  
April 17th, 2013, 05:24 AM   #3
Newbie
 
Joined: Feb 2013

Posts: 15
Thanks: 0

Re: Strange analytic function

Yes. Sorry I guess I should have wrote it like this:

infinity
E
n = 1
bach71 is offline  
April 17th, 2013, 12:26 PM   #4
Global Moderator
 
Joined: May 2007

Posts: 6,834
Thanks: 733

Re: Strange analytic function

Function is analytic since the power series always converges, bounded by r0 + 9e^x. If r is an integer, then you have only r0. If r is rational, you have either a finite number of non-zero decimal places (polynomial) or repeated decimal (I can't describe the function).
mathman is offline  
Reply

  My Math Forum > College Math Forum > Calculus

Tags
analytic, function, strange



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Show function is not analytic WWRtelescoping Complex Analysis 1 February 25th, 2014 07:53 PM
What is an analytic function aaron-math Complex Analysis 4 October 14th, 2013 04:44 PM
Existance of analytic function at C - {0} vitalik.t Complex Analysis 1 January 9th, 2011 10:08 AM
sequence, analytic function eskimo343 Complex Analysis 1 April 19th, 2010 12:53 PM
analytic function krackwacker Complex Analysis 2 November 6th, 2008 10:19 PM





Copyright © 2019 My Math Forum. All rights reserved.