i*****y
发帖数: 8 | 1 I know a laplace transform pair like:
1/[Sqrt(s+a) + Sqrt(s+b)] <-> [Exp(-bt) - Exp(-at))/2(b-a)Sqrt(Pi t^3)]
can anybody tell me how to derive the inverse transform?
thanks | y**t
发帖数: 50 | 2 from left to right, inverse L.T., choose carefully the curve consisting
of a vertical line, a near semicircle (towards left) and two near semilines
(one end is -infinity) just below and above the real axis
with 2 tiny semicircles and a tiny circle around -a and -b. Now the inverse
L.T. becomes 4 integrals, combine them together and you reach the rhs.
from right to left, L.T., easy done by variable substitution.
【在 i*****y 的大作中提到】
: I know a laplace transform pair like:
: 1/[Sqrt(s+a) + Sqrt(s+b)] <-> [Exp(-bt) - Exp(-at))/2(b-a)Sqrt(Pi t^3)]
: can anybody tell me how to derive the inverse transform?
: thanks
| i*****y
发帖数: 8 | 3 The laplace pair should be:
1/[Sqrt(s+a) + Sqrt(s+b)] <-> [Exp(-bt) - Exp(-at)]/[2(b-a)Sqrt(Pi t^3)]
I am still unclear. centainly s = -a or s = -b are not the
singular points of 1/[Sqrt(s+a) + Sqrt(s+b)], how can we
integrate around these two points?
【在 y**t 的大作中提到】
: from left to right, inverse L.T., choose carefully the curve consisting
: of a vertical line, a near semicircle (towards left) and two near semilines
: (one end is -infinity) just below and above the real axis
: with 2 tiny semicircles and a tiny circle around -a and -b. Now the inverse
: L.T. becomes 4 integrals, combine them together and you reach the rhs.
: from right to left, L.T., easy done by variable substitution.
| y**t
发帖数: 50 | 4 The function 1/[Sqrt(s+a) + Sqrt(s+b)] only has singularity on real axis. We
integrate around
these points because Sqrt has a branch cut. If s=-a or s=-b, the angular part
is not well defined.
We use circles to integrate around them. Check some complex analysis by
yourself.
semilines
inverse
【在 i*****y 的大作中提到】
: The laplace pair should be:
: 1/[Sqrt(s+a) + Sqrt(s+b)] <-> [Exp(-bt) - Exp(-at)]/[2(b-a)Sqrt(Pi t^3)]
: I am still unclear. centainly s = -a or s = -b are not the
: singular points of 1/[Sqrt(s+a) + Sqrt(s+b)], how can we
: integrate around these two points?
|
|
