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Question: How to solve transcendental equation exactly?

Question:How to solve transcendental equation exactly?

Markiyan Hirnyk 7759 Maple 2016
solve identify fsolve realrange gamma
September 29 2016
0

I mean the root of the equation

GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n)) = 1

belonging to RealRange(Open(1),4). It should be noticed there are solutions outside this interval. Here is my try.

 

``

solve({n > 1, GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n)) = 1, n < 4}, [n])``

[]

 (1)

`in`(which*is*wrong, view*of)

simplify(eval(GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n)), n = (1/2)*sqrt(5)+1/2))

1

 (2)

Also

Student[Calculus1]:-Roots(A = 1, n = 1 .. 4)

[1.618033989]

 (3)

There is a substitute

fsolve(GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n)) = 1, n = 1 .. 4)

1.618033989

 (4)

NULL

identify(%)

(1/2)*5^(1/2)+1/2

 (5)

``

There is a shade of hope that GAMMA(n-1/n)*GAMMA(1/n)/(n*GAMMA(n))  can be simplified.

Download solution.mw

 PS. An SCR was submitted by me.
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