By Euler's formulae eia = cos a + i sin a and substitute in the above equation. 00 U (a, 0) = (cos ax + i sin ax) dx 0- 00 -|x| cos axdx + i e-lxl sin axdx -00 -00 00 cos axdx + 0 -00 00 -|x| e cos axdx -00
By Euler's formulae eia = cos a + i sin a and substitute in the above equation. 00 U (a, 0) = (cos ax + i sin ax) dx 0- 00 -|x| cos axdx + i e-lxl sin axdx -00 -00 00 cos axdx + 0 -00 00 -|x| e cos axdx -00
By Euler's formulae eia = cos a + i sin a and substitute in the above equation. 00 U (a, 0) = (cos ax + i sin ax) dx 0- 00 -|x| cos axdx + i e-lxl sin axdx -00 -00 00 cos axdx + 0 -00 00 -|x| e cos axdx -00
I don't understand why the integral from -infinity to infinity of e^-abs(x)sinaxdx=0. Can you please explain it to me. Thank you
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Math / Bundle: Differential Equations with Boundary-... | In Problems 1-21 and 24-26 use the Fourier integral transforms of ...
: In Problems 1-21 and 24-26 use the Fourier integral transforms of this section to solv...
Take Fourier transform of the above equation and solve as follows.
F {u (x, 0)} = F {e-lkl}
00
U (α, 0)
e-lxl eiax dx
-00
By Euler's formulae eia
= cos a + i sin a and substitute in the above equation.
00
U (a, 0) = / e
(cos ax + i sin ax) dx
- 0
00
00
,-|x|
cos axdx + i
sin axdx
-00
-00
00
- /
e
cos axdx + 0
-00
-|x|
e
cos axdx
-00
The integration function of the above equation is odd function of x.
00
So, U (a, 0) = 2
cos axdx.
So, the equation is expressed as,
U (α, 0)
(3)
1+a?
Substitute t = 0 in equation (2),
U α, 0) -C.
(4)
Equate the equations (3) and (4) as follows.
c =
1+a?
Substitute the value of c in equation (2),
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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