Results 1 to 5 of 5

Math Help - Complex analysis: Find domain of definition and range

  1. #1
    Banned
    Joined
    Feb 2009
    Posts
    15

    Exclamation Complex analysis: Find domain of definition and range

    Find the domain of definition of each function:

    a. f(z)=3z^2+5z+i+1

    b. g(z)=1/z

    c. h(z)=(z+i)/(z^2+1)

    d. q(z)=(2z^2+3)/(|z-1|)

    e. F(z)=e^3z

    f. G(z)=e^z+e^-z

    Describe the range of each function:

    g. f(z)=z+5 for Re z>0

    h. g(z)=z^2 for z in the first quadrant, Re z >=0, Im z >=0

    i. h(z)= 1/z for 0<|z|<=1

    j. p(z)=-2z^3 for z in the quarter-disk |z| < 1, 0<Arg z<pi/2
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by srw899 View Post
    Find the domain of definition of each function:

    a. f(z)=3z^2+5z+i+1

    b. g(z)=1/z

    c. h(z)=(z+i)/(z^2+1)

    d. q(z)=(2z^2+3)/(|z-1|)

    e. F(z)=e^3z

    f. G(z)=e^z+e^-z
    the domains here will be pretty much the same as if z was a real number.

    Describe the range of each function:

    g. f(z)=z+5 for Re z>0

    h. g(z)=z^2 for z in the first quadrant, Re z >=0, Im z >=0
    Hint: when the domain is not restricted, polynomials are onto the whole complex plane (consequence of the Fundamental theorem of Algebra)



    i. h(z)= 1/z for 0<|z|<=1
    consider, regarding the range: what happens when z approaches zero? what happens if it is on the unit disk?


    j. p(z)=-2z^3 for z in the quarter-disk |z| < 1, 0<Arg z<pi/2
    see my comment for (g) and (h)


    now, what can you come up with?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,242
    Thanks
    1795
    jhevon said "the domains here will be pretty much the same as if z was a real number."

    Pretty much. One important difference is that z^2+ 1 is never 0 in the real numbers but is in the complex numbers it is. So in the complex numbers you have to watch out for fractions with z^2+ 1 in the denominator.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by HallsofIvy View Post
    jhevon said "the domains here will be pretty much the same as if z was a real number."

    Pretty much. One important difference is that z^2+ 1 is never 0 in the real numbers but is in the complex numbers it is. So in the complex numbers you have to watch out for fractions with z^2+ 1 in the denominator.
    indeed. i didn't notice part (c), i should have been more explicit. but i did mention the fundamental theorem of algebra, so we know that all polynomials have zeros, and dividing by zero is always a no-no. so we have to watch out if there is any polynomial in the denominator of a rational function.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by Jhevon View Post
    Hint: when the domain is not restricted, polynomials are onto the whole complex plane (consequence of the Fundamental theorem of Algebra)
    Actually non-constant polynomials, but this trivial point was not why I wanted to responded. In fact any entire function (analytic everywhere) has range the whole complex plane (with an exception of only a single point)! This is such a deep, elegant and amazing result from complex analysis! (Picard's Little Theorem)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: February 23rd 2011, 09:46 AM
  2. how to find the Domain definition of Function
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: December 12th 2010, 06:56 AM
  3. Replies: 1
    Last Post: October 2nd 2010, 02:54 PM
  4. Domain and Range of Complex Functions
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 30th 2010, 07:45 PM
  5. Find Range and Domain!?!?!?!?
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: June 5th 2008, 10:14 AM

Search Tags


/mathhelpforum @mathhelpforum