Solutions to a Functional Equation via Stern-Brocot Tree Fractions

order k

0

1

2

3

4

5

solution g for order k

g versus x

g versus y

singularities

vary y or x

1

g

1

(x,y) = x

1

2

(x+1)



x

x+1

+y

+

1

x+y+1

This Demonstration generates solutions of the functional equation

g

k

(x,y)+

g

k

1

x

,

1

y

=1

starting from

g

0

=

x

x+y

. The solutions correspond to Stern–Brocot tree fractions at level

k

. Change

k

at the top for more complicated solutions. Click the "

g

versus

x

" button to see a plot of the computed solution depending on

y

as a parameter, which you can vary with the "vary

y

or

x

" slider. Similarly, click the "

g

versus

y

" button and vary the parameter

x

. Finally, click the "singularities" button to see the points where the computed function is not defined. The color-coding of those points helps identify where the parts of the solution originated.

Mouse over the plot to see the values or equations of the data. The black and red circled points are related to Stern–Brocot tree fractions. The absolute values of the

y

components of the black circled points are the values of all Stern–Brocot tree fractions up to level

k

. The red circled points have a

y

component with an absolute value equal to the Stern–Brocot tree fractions at level

k+1

.