- #1
matematikuvol
- 192
- 0
Homework Statement
Prove
[tex]\sqrt{\frac{2}{\pi}}\int^{\infty}_0x^{-\frac{1}{2}}\cos (xt)dx=t^{-\frac{1}{2}}[/tex]
and use that to solve
[tex]\int^{\infty}_0\cos y^2dy[/tex]
Is this good way to try to prove?
Homework Equations
The Attempt at a Solution
Homework Statement
Multiplicate both sides with [tex]\cos x'tdt[/tex] and integrate from zero to [tex]\infty[/tex]
[tex]\sqrt{\frac{2}{\pi}}\int^{\infty}_0dt\cos (x't)\int^{\infty}_0x^{-\frac{1}{2}}\cos (xt)dx=\int^{\infty}_0dt\cos (x't)t^{-\frac{1}{2}}=\sqrt{\frac{2}{\pi}}\int^{\infty}_0dxx^{-\frac{1}{2}}\int^{\infty}_0dt\cos (x't)\cos (xt)dx[/tex]