# Question:Fourier transforms

## Question:Fourier transforms

Maple 2020

There are discrepancies between Maple's solution of Fourier transforms and the results printed in USA NIST Handbook of Mathematical Functions, page 30

fourier(exp(-a*abs(x))/sqrt(abs(x)),x,s) assuming a>0;
/   /   (1/2)   (1/2)                (1/2)
1   |   |2 2      Pi      signum(s - _U1)
---- |int|-------------------------------------,
2 Pi |   |       /   2    \
|   |       |_U1     |          (1/2)
|   |     a |---- + 1| (s - _U1)
|   |       |  2     |
\   \       \ a      /

\\
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_U1 = -infinity .. infinity||
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//

For this transform of
"exp(-a*abs(x))/sqrt(abs(x))"

the result in the NIST table is
"sqrt(a + sqrt(a^2 + s^2))/sqrt(a^2 + s^2)"

.
fourier(sinh(a*t)/sinh(Pi*t),x,s) assuming a>-Pi, a<Pi;
2 sinh(a t) Pi Dirac(s)
-----------------------
sinh(Pi t)

For this transform of sinh(a*x)/sinh(Pi*x)   the result in the NIST table is
"1/sqrt(2*Pi)"  "sin(a)/(cosh(s) + cos(a))"

fourier(cosh(a*t)/cosh(Pi*t),x,s) assuming a>-Pi, a<Pi;
2 cosh(a t) Pi Dirac(s)
-----------------------
cosh(Pi t)

For this transform of cosh(a*x)/cosh(Pi*x) the result in the NIST table is
"sqrt(2/Pi) cos(a/2)*cosh(s/2)/(cosh(s) + cos(a))"

These disparities are significant, apart from the fact that Maple failed to solve the first example above.

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