Turn on thread page Beta

Laurent's Theorem watch

Announcements
    • Thread Starter
    Offline

    0
    ReputationRep:
    Hi just a bit of help needed here as I don;t know where to start:

    Part (A)
    ----------------------------
    Suppose f(z) = u(x,y) + iv(x,y)\;and\;g(z) = v(x,y) + iu(x,y) are analytic in some domain D. Show that both u and v are constant functions..?

    I guess we have to use the CRE here but not really sure how to approach this..?

    Part (B)
    ----------------------------
    Let f be a holomorphic function on the punctured disk D'(0,R) = \left\{ {z \in C:0 < |z| < R} \right\} where R>0 is fixed. What is the formulae for c_n in the Laurent expansion:
    

f(z) = \sum\limits_{n = - \infty }^\infty {c_n z_n }.

    Using these formulae, prove that if f is bounded on D'(0,R), it has a removable singularity at 0.

    - Well I know that:
    c_n = \frac{1}

{{2\pi i}}\int\limits_{\gamma _r }^{} {\frac{{f(s)}}

{{(s - z_0 )^{n + 1} }}} ds = \frac{{f^{(n)} (z_0 )}}

{{n!}}.
    Any suggestions from here?


    PART (C)
    -------------------
    Find the maximal radius R>0 for which the function 

f(z) = (\sin z)^{ - 1} is holomorphic in D'(0,R) and find the principal part of its Laurent expansion about z_0=0

    ??

    Any help would be greatly appreciated.

    Thanks a lot
    Offline

    15
    ReputationRep:
    (A) Look to show u_x = u_y = 0
    (B) Apply an estimation theorem to that expression for c_n
    (C) Do you know the sinz series?
    • Thread Starter
    Offline

    0
    ReputationRep:
    hmm so for part (A)
    u_x = v_y = -u_x AND
    u_y = -v_x = v_x

    so u and v are constant because u_x = -u_x and -v_x = v_x

    is that correct?
    • Thread Starter
    Offline

    0
    ReputationRep:
    for part (C):
    yes the sin z series is:

       \sin z = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\mbox{ for all } x\!

    what would be the next step please?
    Offline

    18
    ReputationRep:
    Personally, I'd look at sin z / z.
 
 
 

University open days

  • Manchester Metropolitan University
    Postgraduate Open Day Postgraduate
    Wed, 14 Nov '18
  • University of Chester
    Chester campuses Undergraduate
    Wed, 14 Nov '18
  • Anglia Ruskin University
    Ambitious, driven, developing your career & employability? Aspiring in your field, up-skilling after a career break? Then our Postgrad Open Evening is for you. Postgraduate
    Wed, 14 Nov '18
Poll
Should Banksy be put in prison?
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

Equations

Best calculators for A level Maths

Tips on which model to get

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.