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July 4

Inverse Laplace transform

How can I find or prove inverse Laplace transform?

I can't understand to how do that?

What's a proof? — Preceding unsigned comment added by 151.236.179.200 (talk) 09:48, 4 July 2017 (UTC)

Template:Ping Your question is not sufficiently well-defined for us to provide an immediate answer, but the article inverse Laplace transform provides an explicit formula for the inverse.--Jasper Deng (talk) 09:58, 4 July 2017 (UTC)

Irrationality of Pi

I know that pi is irrational, and therefore either the diameter or the circumference of a circle must be irrational. Can they both be irrational though? If I throw pi x sqrt(5) into my calculator I get a number that superficially appears irrational (doesn't noticeably repeat until the calculator runs out of decimal places), but is it truly irrational? Is there a way to check the rationality of a number when it is a ratio or a product of two irrational numbers? 202.155.85.18 (talk) 09:49, 4 July 2017 (UTC)

Sure. Using your example where the diameter is $\sqrt{5}$, here's a quick proof by contradiction that $\pi \sqrt{5}$ is irrational (even transcendental). If it were rational, then so would $5{\pi }^2$. But we know $\pi$ is in fact a transcendental number, so any rational power of it must be irrational, contradicting the assertion that $\pi \sqrt{5}$ is rational. It follows that $\pi \sqrt{5}$ is irrational.--Jasper Deng (talk) 09:53, 4 July 2017 (UTC)

There are some cases which are much harder to check, but this is not one of them. As an example of such a case, it is not known if πe is irrational. Double sharp (talk) 09:55, 4 July 2017 (UTC)

More generally, the product of any nonzero algebraic number with a transcendental number is still transcendental. The proof proceeds by noting that the algebraic numbers are closed under multiplication; if $ab=c$ where a and c are algebraic, b is transcendental, and $a\ne 0$, then $\frac{1}{a}c=b$ would be algebraic, a contradiction.--Jasper Deng (talk) 10:04, 4 July 2017 (UTC)

Here's another proof that the diameter and circumference can both be irrational. Suppose that this was not true. Then for every irrational number x, it would be true that xπ is a rational number. So it would be possible to create a one-to-one correspondence between all the positive irrational numbers and some of the rational numbers. But by Cantor's diagonal argument, we know that the irrational numbers are uncountably infinite, whereas the rational numbers can be shown to be countably infinite; and therefore no such correspondence is possible. Therefore it is possible that the diameter and circumference can both be irrational. --76.71.5.114 (talk) 10:44, 4 July 2017 (UTC)

It may not be constructive but I like arguments like that which show there is a general reason for something. Dmcq (talk) 15:12, 5 July 2017 (UTC)

Laplace transform

Template:Archive top How can I derive inverse Laplace transform for Laplace transform? I need to know the proof. — Preceding unsigned comment added by 151.236.166.130 (talk) 18:04, 4 July 2017 (UTC)

Template:Ping Please limit the number of questions you ask on this subject to one question.--Jasper Deng (talk) 18:26, 4 July 2017 (UTC)