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October 10

Logarithm

There are also some other integral representations of the logarithm that are useful in some situations:

\ln(x)=-\lim _{\epsilon \to 0}\int _{\epsilon }^{\infty }{\frac {dt}{t}}\left(e^{-xt}-e^{-t}\right)

\ln(x)=\int _{0}^{\infty }\,{\frac {dt}{t}}\,\left[\cos(t)-\cos(xt)\right]

The first identity can be verified by showing that it has the same value at $x = 1$ , and the same derivative. The second identity can be proven by writing ${\frac {1}{t}}=\int _{0}^{\infty }\,dq\,e^{-qt}$

and then inserting the Laplace transform of $cos(xt)$ (and $cos(t)$ ).

Hi, this part is unsourced within a FA article. I'm translating it to a different language and currently looking for its source. Can anyone please help me find the source of the given information? 14.169.224.164 (talk) 15:10, 10 October 2020 (UTC)[reply]

The basic integrals can probably be verified from any standard table. As for the "usefulness", that's questionable and really does need clarification. The limit in the first integral is unnecessary (that's exactly how you define such an improper integral. And the weird notation gives me pause. I'd be inclined to simply remove this. –Deacon Vorbis (carbon • videos) 15:31, 10 October 2020 (UTC)[reply]

I'm still looking for the source where the above information was taken. Being able to verify those doesn't take away the fact that it is unsourced in a FA article, which is unacceptable. 14.169.224.164 (talk) 15:42, 10 October 2020 (UTC)[reply]

Yes, this is not a routine calculation. Like DV, my inclination is to simply remove this. Without explanation of what makes this "useful in some situations" (which situations?), this is useless information. --Lambiam 18:55, 10 October 2020 (UTC)[reply]

Oh, and also, I was about to say that both of these follow from a more general argument without any mention of Laplace transforms, and apparently this has a name, and we have a bit of a (not-so-great) article on it already; see Frullani integral. That has a couple sources of possibly varying quality, but I haven't looked at them in detail yet. –Deacon Vorbis (carbon • videos) 19:04, 10 October 2020 (UTC)[reply]

See here. Count Iblis (talk) 19:01, 10 October 2020 (UTC)[reply]

And also here. Count Iblis (talk) 19:06, 10 October 2020 (UTC)[reply]

Thank you everyone! Frullani integral does have some nice applications. 2402:800:4315:2C28:24D7:48D:5EFC:674C (talk) 22:30, 10 October 2020 (UTC)[reply]

Diagonal depth of sofa

I'm a bit surprised I haven't found this in the archives. We want to buy a sofa and wonder if it will fit into our living room. The most problematic point is the living room door itself but fortunately the door is slightly higher than the length of the sofa. The gap through the doorway is about 860 mm wide but the sofa is 950 mm high and 970 mm deep so it won't fit through "orthogonally". Many web pages say it will fit through if the "diagonal depth" is less than the width of the door. Some sites say the length AD (or BD) is the diagonal depth but this is silly. Others say the length of the line through B that reaches the midpoint (or bisects) AD gives the diagonal depth but I suspect this is also wrong. More likely (to me) is the line through B that is at right angles to AD. I can picture the line AD sliding along one edge of the doorway so that point B would need to pass the other edge. My trigonometry on right triangles AED and AFB suggests this length, BF, is about 890 mm which is bad news! But can't we do better than this by sliding the sofa in until the point marked with a red O is at one edge of the door and then rotate about this point so that B just goes past the other edge? But I have no idea where point O would be or even whether it would be on the line BF. If there are two reliable sources I could create Diagonal depth but even if not can anyone help, please? Thincat (talk) 21:14, 10 October 2020 (UTC)[reply]

First, my calculation confirms that AD BF is 890 mm. Second, you're right, this is irrelevant. It would be relevant if you wanted to take one end inside first, with the lower back side of the sofa going in one side of the door, and the upper back and front going in the other side. But what you're talking about is standing the sofa on one end and first putting the back through the door, then rotating to put the base through, or vice versa. Since the opening is "higher than the length of the sofa", standing it on end will get it through the height of the doorway. And as long as there are no other obstructions nearby that would keep you from rotating it, we don't need to calculate an exact position for O; a simple sketch shows that there's plenty of room. --174.89.48.182 (talk) 21:58, 10 October 2020 (UTC)[reply]

Ah, yes, thank you. I can now see that "diagonal depth" is only relevant if you are wanting to slide the thing in lengthways. In many situations that would be necessary when the door is lower than the length of the sofa. That never occurred to me. Thincat (talk) 22:12, 10 October 2020 (UTC)[reply]