How Can I Derive Sumations with Multiple Indices and Powers?

  • Context: Graduate 
  • Thread starter Thread starter jetoso
  • Start date Start date
Click For Summary

Discussion Overview

The discussion revolves around deriving summations involving multiple indices and powers, specifically focusing on the functions P(z) and Q(z) defined as infinite series. Participants explore methods for manipulating these summations to achieve a simplified form.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant presents a complex summation involving indices i and j, seeking advice on how to derive it.
  • Another participant expresses confusion regarding the original question, indicating a lack of clarity in the problem statement.
  • A subsequent reply suggests a simplification of the original summation to a specific form, although the reasoning behind this simplification is not detailed.
  • Another participant proposes that switching the order of summation is a key technique, outlining how this could lead to two distinct parts in the summation.
  • One participant expresses continued confusion, indicating that they do not fully understand the proposed methods.
  • A final post questions the understanding of switching the order of summation and the change of variables, suggesting that these concepts are crucial to resolving the problem.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the clarity of the original problem or the effectiveness of the proposed methods. Multiple viewpoints and levels of understanding are present, indicating ongoing confusion and exploration.

Contextual Notes

There are limitations in the clarity of the problem statement and the assumptions made regarding the summation techniques. The discussion does not resolve the mathematical steps involved in the derivation.

jetoso
Messages
73
Reaction score
0
I have a hard time trying to figure it out how to derive the following sumations:

We know that:
P(z) = Sumation from j=0 to Infinity of [(Pj * (z^j)]
and
Q(z) = Sumation from j=0 to Infinity of [(Kj * ((z^j)]

Where j is a subindex for P and K, and a power for z.


Sumation from j=0 to Infinity of [ Sumation from i=0 to j+1 of [(z^j) * Pi * Kj-i+1] ]

Where i and j are a subindices for P and K, and a power for z.



I know that I have to play with the indices, but I have no clue. Any advice??
 
Physics news on Phys.org
I may be missing something, but I can't see what your question is?
 
Reply

Yes, you are right.
From the information above, it has to be simplified to:
[ (1/z) * P(z) * Q(z) ] - [ (1/z) * Po * Q(z) ]

The author of the book does not give much more information about how he derived this equation, but he says is trivial.
 
The basic trick is to switch the order of summation. Then you will have two parts. The main part has i=1,inf with j=i-1,inf. The other part has i=0 with j=0,inf. For the main part let n=j-i+1 (replacing j), then n=0,inf. Put it all together and you should get the required answer.
 
Sorry

I am sorry, but I still do not get it.
 
Do you understand switching the order of summation?
Do you see the result of switching?
Do you understand the change of variables (n=)?
 

Similar threads

Graduate How do I solve y*exp(y)=[constant] for J using the Lambert W function?
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Intersections between this infinite power tower and a multifactorial
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad I would like to know the calculation process of this power spectrum
  • · Replies 8 ·
Replies
8
Views
2K
Graduate Is the Chain Rule Applicable to the Euclidean Norm in Calculating Derivatives?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Help needed with derivation: solving a complex double integral
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Existence of a limit implies that a function can be harmonic extended
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Anyone can have the idea how to solve the log of a power ser
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Expansion of a complex function around branch point
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Question about proof of Great Picard Theorem
  • · Replies 3 ·
Replies
3
Views
3K
Graduate The partial derivative of a function that includes step functions
  • · Replies 3 ·
Replies
3
Views
4K