Join TSR now and get all your revision questions answeredSign up now

FP2: Complex numbers problem. Watch

    • Thread Starter
    Online

    3
    ReputationRep:
    Hi, I've been set a homework sheet and think I've made a mistake somewhere along this question:

    Show that 1 + e^(jx) = (2cos((1/2)x))e^((1/2)jx)

    My working thus far (may have mistakes in, I'm tired and braindead):

    1 + e^(jx)
    = 1 + cosx + jsinx
    = 1 + (1/2)(2cos^2((1/2)x) + 1) + 2jsin((1/2)x)cos((1/2)x)
    = (3/2) + cos^2((1/2)x) + 2jsin((1/2)x)cos((1/2)x)
    = (3/2) + (2cos((1/2)x)(1/2cos((1/2)x) + jsin((1/2)x))

    And naturally I would turn that last bit into e^((1/2)jx) if I had a whole cos((1/2)x), but it still wouldn't be the result I want.

    Sorry if this is hard to read, but I'm hopeless with LaTeX.

    Cheers.
    Offline

    2
    ReputationRep:
    On this line:

    (Original post by TheDefiniteArticle)
    = 1 + (1/2)(2cos^2((1/2)x) + 1) + 2jsin((1/2)x)cos((1/2)x)
    The 1/2 shouldn't be there, and the +1 should be -1.

    You should find that things cancel/simplify from here.
    • Thread Starter
    Online

    3
    ReputationRep:
    Well aren't I a daft ****.

    Thanks.
    Offline

    2
    (Original post by TheDefiniteArticle)
    Show that \displaystyle 1+e^{jx}=2e^{\frac{jx}{2}} cos \left( \frac{x}{2} \right).
    I personally would have started with the RHS, using the following identity. 2cos(x)=\frac{e^{jx}+e^{-jx}}{2}

    Aside: This can be shown using the fact that e^{jx}=cos(x)+jsin(x).

    It should make life a lot easier.

    I hope it helps.

    Darren
 
 
 
Poll
Is GoT overrated?
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

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