Turn on thread page Beta

sum of complex exponentials result-zero/non-zero watch

    • Thread Starter
    Offline

    12
    ReputationRep:
    Hi,

    This is probably a stupid question, I understand that it is equal to p if p divides n.
    But I don't understand why it is zero otherwise.

    For example if I have n/p=1/2
    then

     \sum\limits^{p-1}_{j=0} e^{\frac{2\pi i j}{2}} = \sum\limits^{p-1}_{j=0} e^{\pi i j}

    E.g also say p=5 then I have

     = 1+e^{\pi i  } + e^{2\pi i }  + e^{3\pi i }  + e^{4\pi i } =1-1+1-1+1

    i gues im doing something stupid?

    ta.
    Attached Images
     
    Offline

    15
    ReputationRep:
    (Original post by nologiceretoday)
    Hi,

    This is probably a stupid question, I understand that it is equal to p if p divides n.
    But I don't understand why it is zero otherwise.

    For example if I have n/p=1/2
    then

     \sum\limits^{p-1}_{j=0} e^{\frac{2\pi i j}{2}} = \sum\limits^{p-1}_{j=0} e^{\pi i j}

    E.g also say p=5 then I have

     = 1+e^{\pi i  } + e^{2\pi i }  + e^{3\pi i }  + e^{4\pi i } =1-1+1-1+1

    i gues im doing something stupid?

    ta.
    It is a geometric series so just use the formula and the result just falls out.
    • Study Helper
    Offline

    15
    Study Helper
    (Original post by nologiceretoday)
    Hi,

    This is probably a stupid question, I understand that it is equal to p if p divides n.
    But I don't understand why it is zero otherwise.

    For example if I have n/p=1/2
    then

     \sum\limits^{p-1}_{j=0} e^{\frac{2\pi i j}{2}} = \sum\limits^{p-1}_{j=0} e^{\pi i j}

    E.g also say p=5 then I have

     = 1+e^{\pi i  } + e^{2\pi i }  + e^{3\pi i }  + e^{4\pi i } =1-1+1-1+1

    i gues im doing something stupid?

    ta.
    If n/p is 1/2, and p is also 5, then n = 2.5, and isn't an integer.

    B_9710's post covers the part about equalling zero.
    See DFranklin, below
    • Thread Starter
    Offline

    12
    ReputationRep:
    (Original post by B_9710)
    It is a geometric series so just use the formula and the result just falls out.
     r=e^{\frac{2\pi i n}{p}}  s_{p-1}= \sum\limits^{p-1}_{j=0} r^j= \frac{1-e^{2\pi i n}}{1-e^{\frac{2\pi i n}{p}}}  =0 by the numerator. Now if I am to consider the case that p divides n, and rename this integer as m I have  s_{p-1}= \frac{1-e^{2\pi i mp}}{1-e^{2\pi i m}} how would I evaluate this to p?
    Offline

    18
    ReputationRep:
    (Original post by nologiceretoday)
     r=e^{\frac{2\pi i n}{p}}  s_{p-1}= \sum\limits^{p-1}_{j=0} r^j= \frac{1-e^{2\pi i n}}{1-e^{\frac{2\pi i n}{p}}}  =0 by the numerator. Now if I am to consider the case that p divides n, and rename this integer as m I have  s_{p-1}= \frac{1-e^{2\pi i mp}}{1-e^{2\pi i m}} how would I evaluate this to p?
    Note that the standard formula for the sum of a GP is not valid when the common ratio is 1. On the other hand, it is not hard to find the sum of a series of p terms that are all the same.
 
 
 

University open days

  1. University of Bradford
    University-wide Postgraduate
    Wed, 25 Jul '18
  2. University of Buckingham
    Psychology Taster Tutorial Undergraduate
    Wed, 25 Jul '18
  3. Bournemouth University
    Clearing Campus Visit Undergraduate
    Wed, 1 Aug '18
Poll
How are you feeling in the run-up to Results Day 2018?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.