Triple Product in Laplace Transform

In summary, this problem appears to be much more difficult than I thought it would be, and I don't think I will be able to solve it using integration by parts.
  • #1
echandler
21
1
Hello - I'm not sure this is where this should go, but I'm working with Laplace Transforms and differential equations, so this seems as good a place as any. Also, I doubt this is graduate level math strictly speaking, but I went about as high as you can go in calculus and linear algebra during my undergraduate degree, and wouldn't have known where to begin with some of the stuff in this problem.

So I have the equation Int[(x^2)*(e^-sx)*f'(x)dx] on (0,infinity), typical Laplace Transform, except that there is a triple product integration by parts. After hours of integrating by parts and getting nowhere, I made a map of the possible outcomes, shown in the pictures below, and even before asking my teacher if there was some method of figuring out from the map if it had a possible solution by Integration by parts, to which he said he had no idea where to begin with answering that question, I was pretty confident that a solution via integration by parts is not possible. So I though, well, why not write a solution in the form of an infinite series, combine the x^2 term with the series form of e^x term, then, put it back in the integral with f'(x) and now we are back with integration by parts of two functions. Brilliant.

But I have a question. I'm not a mathematician, so forgive me if this is really basic, but if x^2 can be represented by the series: 0*(x^0) + 0*(x^1) + 1*(x^2) + 0*(x^3) + 0*(x^4) + ..., then it can be represented as Sigma[a_n*(x^n)] where n goes from (0,infinity), where a_n = 0 for all n except n=2, where a = 1, then it seems that you can use the Product rule for summations,
text44.gif
, (taken from http://functions.wolfram.com/GeneralIdentities/12/), which to me looks like just a scalar product of two infinite series, and you end up getting only the x^2 terms from each respective series out from the result, because all the other terms in the (x^2) "expansion" are equal to zero and thus "kill" the corresponding terms in the series for the exponential, meaning: (x^2)*exp(x) = (x^4)/2....But this can't really be, can it? I am almost positive I'm missing a fundamental piece of understanding, but I can't seem to determine where I'm going wrong. Please tell me what piece of understanding about series that I am missing!

Thank you in advance.
 

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  • #2
echandler said:
and you end up getting only the x^2 terms from each respective series out from the result

This is not correct. When you have a double summation you have to iterate through each index independently.

For example the sum
[itex] \sum_i \sum_j a_i b_j = a_0b_0 + a_0 b_1 + a_0 b_2 + \dots + a_1 b_0 + a_1 b_1 + a_1 b_2 + \dots + a_2 b_0 + a_2 b_1 + a_2 b_2 + \dots [/itex]

In the case where t [itex] a_2 = x^2 [/itex] and all other [itex] a_i = 0 [/itex] the sum is
[itex] \sum_i \sum_j a_i b_j = 0 + x^2b_0 + x^2 b_1 + x^2 b_2 + \dots = x^2 \sum_j b_j [/itex].
 
  • #3
Oh wow. That makes so much sense.

Thanks so much!
 

Related to Triple Product in Laplace Transform

What is the triple product in Laplace Transform?

The triple product in Laplace Transform is a mathematical formula used to simplify the calculation of certain integrals involving three different variables. It is commonly used in engineering and physics to solve problems involving complex systems.

How is the triple product in Laplace Transform calculated?

The triple product in Laplace Transform is calculated by taking the product of three different functions, each of which contains one of the three variables in the integral. This product is then integrated with respect to all three variables, resulting in a simplified expression.

What is the significance of the triple product in Laplace Transform?

The triple product in Laplace Transform is significant because it allows for the simplification of complex integrals, making them easier to solve and analyze. This is particularly useful in applications where multiple variables are involved, such as in engineering and physics problems.

What are some examples of where the triple product in Laplace Transform is used?

The triple product in Laplace Transform is commonly used in the analysis of electrical circuits, mechanical systems, and control systems. It can also be applied to problems in fluid dynamics, heat transfer, and other areas of science and engineering.

Are there any limitations to using the triple product in Laplace Transform?

While the triple product in Laplace Transform is a useful tool for simplifying integrals, it may not always be applicable or appropriate for every problem. In some cases, alternative methods may be more suitable for solving the integral. Additionally, care must be taken when using the triple product to ensure that the resulting expression accurately represents the original integral.

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