Continued Fraction for Tan(x)

Date: 11/03/2000 at 09:56:45
From: Joseph 
Subject: "Continued Fraction" for the Tangent Function

Dear Dr. Math,

I was reading about pi on a math site. I saw something that blew my 
mind! They said that the first argument that pi has to be irrational 
was based on this formula (I've copied it down so I can type it out 
exactly):

                          1
     tan x = ----------------------------
              1              1
              - + -----------------------
              x    3           1
                   - + ------------------
                   x    5         1
                        - + -------------
                        x    7      1
                             - + --------
                             x    9
                                  - + ...
                                  x

Can you give me a hint as to why this formula is true? They called it 
an "infinite continued fraction representation." I know about 
convergent series and infinite products, so I think I understand how 
you would define this. But how on earth did they get this particular 
formula? I can follow the rest of the argument from there. Thanks!

Date: 11/03/2000 at 11:24:36
From: Doctor Rob
Subject: Re: "Continued Fraction" for the Tangent Function

Thanks for writing to Ask Dr. Math, Joseph.

The above formula is erroneous. You have written the continued 
fraction for tanh(x), the hyperbolic tangent of x. Correct is:

                          1
     tan x = ----------------------------
              1              1
              - - -----------------------
              x    3           1
                   - - ------------------
                   x    5         1
                        - - -------------
                        x    7      1
                             - - --------
                             x    9
                                  - - ...
                                  x

This can be rewritten in the form

                          x
     tan x = -----------------------------
                            x^2
              1 - ------------------------
                              x^2
                  3 - --------------------
                                x^2
                      5 - ----------------
                                  x^2
                          7 - ------------
                                    x^2
                              9 - --------

                                  11 - ...

Define

                infinity
     Psi(c;z) =   SUM    1/[c*(c+1)*...*(c+k-1)] * z^k/k!
                  k=0

where c is a complex constant but not a nonpositive integer, and z is 
a complex variable. Sometimes this is denoted by 0F1(c;z). Observe 
that

     sin(x) = x*Psi(3/2;-x^2/4)
     cos(x) = Psi(1/2;-x^2/4)

Let {a(n)} be a sequence of complex numbers defined by

     a(n) = 1/[(c+n-1)*(c+n)],  n = 1, 2, 3, ...
Let
     P(n) = Psi(c+n;z),  n = 0, 1, 2, 3, ...

Then it is easily shown that the power series P(n) satisfies the 
3-term recurrence relation

     P(n) = P(n+1) + a(n+1)*z*P(n+2),  n = 0, 1, 2, ...

Divide both sides by P(n+1), and rearrange to get

     P(n)/P(n+1) = 1 + a(n+1)*z/[P(n+1)/P(n+2)]

Thus, starting with n = 0 and doing chain substitution, you'll see 
that

                              a(1)*z
     P(0)/P(1) = 1 + ------------------------
                                a(2)*z
                     1 + --------------------
                                  a(3)*z
                         1 + ----------------
                                    a(4)*z
                             1 + ------------
                                      a(5)*z
                                 1 + --------

                                     1 + ...
  
I leave the matter of convergence aside, but it does work for all 
complex numbers z. Now when c = 1/2, P(0)/P(1) is just

     P(0)/P(1) = Psi(1/2;z)/Psi(3/2;z)
and
     a(n) = 4/[(2*n-1)*(2*n+1)]

so you get the equation
  
                                         4*z/(1*3)
     Psi(1/2;z)/Psi(3/2;z) = 1 + --------------------------
                                           4*z/(3*5)
                                 1 + ----------------------
                                             4*z/(5*7)
                                     1 + ------------------
                                               4*z/(7*9)
                                         1 + --------------
                                                 4*z/(9*11)
                                             1 + ----------

                                                 1 + ...

Now substitute z = -x^2/4, and you get

                                     -x^2/(1*3)
     cos(x)/[sin(x)/x] = 1 + ---------------------------
                                       -x^2/(3*5)
                             1 + -----------------------
                                         -x^2/(5*7)
                                 1 + -------------------
                                           -x^2/(7*9)
                                     1 + ---------------
                                             -x^2/(9*11)
                                         1 + -----------
  
                                             1 + ...
  
From here it is not too difficult a rearrangement to get the continued 
fraction first mentioned in my answer above.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/