Volume 10, Issue 2

Articles
In and Out
Trott's Corner
Beyond Sudoku
New Products
New Publications
Calendar
News Bulletins
New Resources
Classifieds

Editorial Policy
Staff and Contributors
Submissions
Subscriptions
Back Issues
Contact Information

In and Out

Q: How can I solve the functional equation ?

A: The solution to this functional equation is given at mathworld.wolfram.com/FunctionalEquation.html. Noting that

then taking logs of both sides, one sees that , where is arbitrary, satisfies the functional equation. More generally, since

we observe that is a solution to the functional equation

for .

One way to solve the functional equation is to assume that the asymptotic behavior of the solution is .

Substitute this sum into the functional equation and expand into an asymptotic series.

Solving for the coefficients , one sees that .

Summing the asymptotic series, one obtains the same solution as earlier, .

Here we apply the same method to .

The pattern of the coefficients is clear: the coefficient is . Summing the asymptotic series, one obtains the general solution.

Absorbing the factor into the arbitrary constant, the solution can be written as . This solution results by taking the logarithm of the following identity.

Finally, we apply the same method to .

The pattern of the coefficients is clear: the even coefficients vanish, , and the odd coefficients read . Summing the asymptotic series, one obtains the general solution.