Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Taylor Series in Multiple Variables

  1. Sep 4, 2010 #1
    Can anyone help me for the leading order terms in the taylor series for the function
    f(x,y) = Sqrt(a*x^8+b*x^4*y^4+y^8),
    centered at x=0,y = 0 and a,b,c constants?
     
  2. jcsd
  3. Sep 8, 2010 #2
    Your question really depends on the values of x and y. And your expression doesn't have c. I assumed your function to be:

    [tex]f(x,y)=\sqrt{a x^8+b x^4y^4+c y^8}[/tex]

    I only outline the method to obtain a power series here.

    [tex]f(x,y)=c y^8\sqrt{1+\left(\frac{a x^8+b x^4y^4}{c y^8}\right)}[/tex]

    By

    [tex](1+x)^p=\sum _{k=0}^{\infty } \binom{p}{k}x^k[/tex]

    ,we have:

    [tex]f(x,y)=c y^8\sum _{k=0}^{\infty } \binom{\frac{1}{2}}{k}\left(\frac{a x^8+b x^4 y^4}{c y^8}\right)^k[/tex]

    There is another expansion method as well, in which you take out the power x instead of y. It depends on the values of x and y. Since the expansion of [tex](1+x)^p[/tex] is valid for |x| < 1 (|x| = 1 is more complicated).
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Taylor Series in Multiple Variables
  1. Taylor Series (Replies: 4)

  2. Taylor series (Replies: 2)

  3. Taylor series (Replies: 7)

  4. Taylor series (Replies: 5)

Loading...