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Volume 10, Issue 2

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Edited by Paul Abbott

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.

See also [3], [4], and [5].



     
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