Results 1 to 2 of 2

Math Help - counter integral

  1. #1
    Senior Member
    Joined
    Jan 2008
    From
    Montreal
    Posts
    311
    Awards
    1

    counter integral

    Let y(x) be a real valued function defined on the interval 0<x\leq 1 by means of the equations:

    y(x) = \left\{ \begin{array}{rcl}<br />
x^3 \sin(\pi/x) & \mbox{for} & 0<x\leq 1 \\ <br />
0 & \mbox{for} & x=0<br />
\end{array}\right.<br />

    show that the equation z=x+iy(x) \ (0\leq x \leq 1) representing an arc C that intersects the real axis at the point z=\frac{1}{n} \ (n=1,\ 2\, \ \dotso) and z=0

    I'm not sure if I'm picking the right values for this, but I would think that it would something along the lines of:

    \int_a^b f[z(t)]z^,(t) dt

    =\int_0^1 \bigg{(} x+i(x^3 \sin(\pi/x)) \bigg{)}\cdot\left(\frac{d}{dx} x^3 \sin(\pi/x)) \right) dx this is the part that I'm not sure of, but continuing would give me:

    \int_0^1 \bigg{(} x+i(x^3 \sin(\pi/x)) \bigg{)} \cdot \bigg{(}3x^2\sin(\pi/x)-x\cos(\pi/x)\cdot\pi \bigg{)} dx

    \int_0^1 3x^3\sin(\pi/x)-x^2\pi\cos(\pi/x)+i(3x^5\sin^2(\pi/x)-x^4\pi\sin(pi/x)\cos(\pi/x)) \ dx

    putting the above equation into Maple gave me:

    -\frac{1}{48} \pi^5 + \frac{1}{24} \cdot Si(\pi) \pi^4 - \frac{1}{24} \pi^3+\frac{1}{12}\pi for some reason this doesn't look correct.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Aug 2008
    Posts
    120
    Your method of attack seems very interesting but is unnecessary.
    Note that for x=0
    z=0
    which is one of the intersections.
    Now for x>0 we have
    z=x+x^3sin(\frac{\pi}{x})i.
    This intersects the real axis when the imaginary component is zero. This happens when
    x^3sin(\frac{\pi}{x})=0.

    Now solve for x.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. when is a counter example not enough?
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: September 20th 2011, 11:34 AM
  2. counter example
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 1st 2009, 11:07 PM
  3. Asynchronous up counter
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: September 25th 2009, 08:12 AM
  4. I need a counter example
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: April 13th 2009, 01:34 AM
  5. Counter-intuitive? or...
    Posted in the Algebra Forum
    Replies: 5
    Last Post: April 3rd 2007, 08:43 PM

Search Tags


/mathhelpforum @mathhelpforum