Bounding e raised to a polynomial - Tell me if this is true?

  • Thread starter Skatch
  • Start date
18
0
Can someone tell me if the following statement is true?

Say you have [tex]P(x)e^{Q(x)}[/tex] where P(x) is any finite-degree polynomial and Q(x) is a polynomial of integer order k. Is it true that for any positive epsilon, I can find some real numbers A and B such that

[tex]|P(x)e^{Q(x)}| \leq Ae^{B|x|^{k+\epsilon}}\quad \forall x ?[/tex]

I don't need a proof, just some intuition about why its true or a counterexample if its not. I figure since the expression on the right will be growing faster than that on the left, I should be able to find some A and B to ensure the RHS will always be larger, right? Is this an obvious statement or am I crazy??

Edit: I think its true since the smallest the right hand side can be is A at x=0. So I choose A so that the RHS is larger at x=0, and B is chosen to ensure that its larger around zero, and then far from zero the growth of the exponent on the RHS will dominate and we are home free... This is the idea, yes?
 
Last edited:

mathman

Science Advisor
7,702
392
Intuition:
|P(x)| < Ce|x|ε for some C and ε > 0. Similarly |Q(x)| < A|x|k for some A. Put it all together to get what you want.
 
Last edited:

Related Threads for: Bounding e raised to a polynomial - Tell me if this is true?

Replies
1
Views
1K
Replies
4
Views
1K
Replies
6
Views
5K
  • Posted
Replies
6
Views
14K
Replies
2
Views
2K
Replies
26
Views
17K
Replies
7
Views
2K
  • Posted
Replies
2
Views
7K
Top