Extended tables of Laplace transforms

[Muddyrunner]

I'm looking for "extended" tables of Laplace transforms i.e. ones which have examples beyond the basics commonly shown in tables. I have already linked to this one in another thread:

http://www.me.unm.edu/~starr/teaching/me380/Laplace.pdf

which is a good indicator of what I am looking for. Any more like this out there that people know about? I have tried Googling, of course, but the number of hits returned is huge.

Regards,

MR

 

[AlephZero]

Try here: http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm

The Laplace transform is a mathematical technique used in engineering, physics, and mathematics to simplify the analysis of linear time-invariant systems. It converts a function of time into a function of a complex variable, s. Here's a basic Laplace transform table that provides some common transforms:

1. Basic Functions:
   • Laplace Transform of a constant:L{1} = 1/s
   • Laplace Transform of a time delay, e.g., e^(-at)u(t):L{e^(-at)u(t)} = 1 / (s + a), where u(t) is the unit step function.
   • Laplace Transform of a sine or cosine function:L{sin(ωt)} = ω / (s^2 + ω^2)L{cos(ωt)} = s / (s^2 + ω^2)
2. Time Scaling:
   • If F(s) is the Laplace transform of f(t), then L{f(at)} = (1/a)F(s/a)
3. Linearity:
   • If F1(s) and F2(s) are the Laplace transforms of f1(t) and f2(t), then L{af1(t) + bf2(t)} = aF1(s) + bF2(s)
4. Derivatives:
   • Laplace Transform of the derivative of f(t):L{f'(t)} = sF(s) - f(0)
   • Laplace Transform of the n-th derivative of f(t):L{f^(n)(t)} = s^nF(s) - s^(n-1)f(0) - s^(n-2)f'(0) - ... - f^(n-1)(0)
5. Integration:
   • Laplace Transform of the integral of f(t):L{∫[0 to t] f(τ) dτ} = 1/s F(s)
6. Unit Step Function:
   • Laplace Transform of the unit step function, u(t):L{u(t)} = 1/s
7. Dirac Delta Function:
   • Laplace Transform of the Dirac delta function, δ(t):L{δ(t)} = 1
8. Exponential Decay:
   • Laplace Transform of e^(-at)u(t):L{e^(-at)u(t)} = 1 / (s + a)
9. Shift Theorem:
   • If F(s) is the Laplace transform of f(t), then L{e^(at)f(t)} = F(s - a)
10. Convolution Theorem:

• Laplace Transform of the convolution of two functions f(t) and g(t):L{f(t) * g(t)} = F(s)G(s), where * represents convolution.

Please note that this is not an exhaustive table, and there are many more Laplace transforms and properties that can be useful for solving various problems in mathematics and engineering. The Laplace transform is a powerful tool for analyzing linear time-invariant systems, differential equations, and other mathematical and scientific problems.

 

[Muddyrunner]

Thanks, AlephZero - some there I had never seen before.