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Dec
8
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Dec
1
revised Function such that $f(x) f(\pi/2 - x) = 1$
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Dec
1
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Dec
1
accepted Function such that $f(x) f(\pi/2 - x) = 1$
Dec
1
revised Function such that $f(x) f(\pi/2 - x) = 1$
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Dec
1
comment Function such that $f(x) f(\pi/2 - x) = 1$
@StevenStadnicki okay, I see. However, I have to choose the $C^\infty$ function such that its power series works at the midpoint. Setting all the derivatives to zero is like making it locally the constant function, and I could also locally make it match a $\tan$ function. Maybe I should have asked for how many analytic functions satisfy the requirement, since each analytic function would characterize the type of power series the $C^\infty$ function can have at the midpoint.
Dec
1
revised Function such that $f(x) f(\pi/2 - x) = 1$
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Dec
1
comment Function such that $f(x) f(\pi/2 - x) = 1$
by smooth I mean $C^\infty$. Looking at the second derivative at $x=pi/4$ (or I guess $1$ in your example) I think gives a nontrivial restriction on the function at that point, $f''-(f')^2=0$. Higher derivatives give still more restrictions, and I'm wondering if $tan(x)$ and the constant function are the unique solutions satisfying all these smoothness conditions.
Nov
30
comment Function such that $f(x) f(\pi/2 - x) = 1$
I changed the interval to $0<x<\pi/2$, I think it makes more sense that way.
Nov
30
revised Function such that $f(x) f(\pi/2 - x) = 1$
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Nov
30
revised Function such that $f(x) f(\pi/2 - x) = 1$
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Nov
30
comment Function such that $f(x) f(\pi/2 - x) = 1$
@StevenStadnicki Are you sure this implies smoothness of the function at $\pi/4$? I'm not seeing it immediately...
Nov
30
comment Function such that $f(x) f(\pi/2 - x) = 1$
@KristofferRyhl oh whoops, I should probably say that it doesn't have to hold at $x=0$ since I want to allow $f$ to diverge as $x\rightarrow \pi/2$.
Nov
30
awarded  Student
Nov
30
asked Function such that $f(x) f(\pi/2 - x) = 1$
Oct
3
answered Mobius Transformation
Feb
16
revised Vector Calculus Identities Using Differential Forms
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Feb
14
revised Vector Calculus Identities Using Differential Forms
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Feb
14
answered Vector Calculus Identities Using Differential Forms
Oct
8
awarded  Yearling