Input

ℱ_x[1/sqrt(x)](s)

Result

1/sqrt(2 π) integral_(-∞)^∞ 1/sqrt(x) e^(i s x) dx = -((1/2 - i/2) (sgn(s) - 1))/sqrt(abs(s))

Plot

Plot
Plot

Alternate form

((1/2 - i/2) - (1/2 - i/2) sgn(s))/sqrt(abs(s))

Expanded form

(1/2 - i/2)/sqrt(abs(s)) - ((1/2 - i/2) sgn(s))/sqrt(abs(s))
(-1/2 + i/2) (sgn(s)/sqrt(abs(s)) - 1/sqrt(abs(s)))

Alternate form assuming s>0

0

Alternate form assuming s is real

(1 - sgn(s))/(2 (s^2)^(1/4)) + (i (sgn(s) - 1))/(2 (s^2)^(1/4))

Trigonometric transforms

ℱ_x[1/sqrt(x)](s) = ℱ_x^c[1/2 (1/sqrt(-x) + 1/sqrt(x))](s) + i (ℱ_x^s[(sqrt(-x) + sqrt(x))/(2 x)](s))

Fourier cosine transform for the even part

sqrt(2/π) integral_0^∞ 1/2 (1/sqrt(-x) + 1/sqrt(x)) cos(s x) dx = (1/2 - i/2)/sqrt(abs(s))

Fourier sine transform for the odd part

sqrt(2/π) integral_0^∞ ((sqrt(-x) + sqrt(x)) sin(s x))/(2 x) dx = ((1/2 + i/2) sgn(s))/sqrt(abs(s))
POWERED BY THE WOLFRAM LANGUAGE