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March 16

Series for pi

Machin-like_formula says:

—

The following equations were derived in Angle addition formula:

$\sin(a_{1}+b_{1})=\sin a_{1}\cos b_{1}+\cos a_{1}\sin b_{1}$

$\cos(a_{2}+b_{2})=\cos a_{2}\cos b_{2}-\sin a_{2}\sin b_{2}$

Algebraic manipulation of these equations yields the following:

$\arctan {\frac {a_{1}}{b_{1}}}+\arctan {\frac {a_{2}}{b_{2}}}=\arctan {\frac {a_{1}b_{2}+a_{2}b_{1}}{b_{1}b_{2}-a_{1}a_{2}}},$ (3)

—

Could someone explain what the manipulation is. -- SGBailey (talk) 13:17, 16 March 2022 (UTC)

Working backwards, the sequence of steps is as follows: The arctan equation is a case of the Arctangent addition formula, which, as explained there, can be derived from the formula for $\tan(x+y)$. And that can be derived from the starting formulas via $\tan =\sin /\cos$. Is that sufficient? --Wrongfilter (talk) 13:55, 16 March 2022 (UTC)

Wow, that is a confusing presentation. The variables $a_{i},b_{i}$ in the $\arctan$ formula do not correspond to the synonymous variables in the angle addition formulas for $\sin$ & $\cos$. And why not directly appeal to the arctangent addition formula? Then this is immediate, apart from some elementary algebraic simplification of a compound fraction.  --Lambiam 15:47, 16 March 2022 (UTC)

I have streamlined the presentation (as well as the formatting) in that section. --JBL (talk) 16:08, 16 March 2022 (UTC)

Thanks - now I understand where 5 and 239 mathemagically came from. -- SGBailey (talk) 09:29, 17 March 2022 (UTC)

March 19

Dissemination problem

Consider $n$ people, each of whom knows a particular fact. How many two-person meetings are required for everybody to know everything, where at each meeting everything known to date is shared? I can't get below $2n-3$, achieved in this way, for example: person $1$ meets everyone else up to person $n$ (taking $n-1$ meetings), then re-meets everyone from $2$ to $n-1$ (taking $n-2$ meetings). Is it possible to do the job in fewer, or indeed to prove that $2n-3$ is the lowest number? →2A00:23C6:AA07:4C00:4C7E:441:A97:15A2 (talk) 13:30, 19 March 2022 (UTC)

It is certainly possible to do better in general: with $n=4$, for example, the meetings AB, CD, AC, BD work. But in fact you only get that one bit of improvement: see https://oeis.org/A058992 . -- JBL (talk) 15:47, 19 March 2022 (UTC)

March 21

Change of variables formula

In the article Integration by substitution the following change of variables formula is mentioned:

Let φ : [a, b] → I be a differentiable function with a continuous derivative, where I ⊆ R is an interval. Suppose that f : I → R is a continuous function. Then

$\int _{a}^{b}f(\varphi (x))\varphi '(x)\,dx=\int _{\varphi (a)}^{\varphi (b)}f(u)\,du.$

My question is how to apply this is the measure theory sense without $\phi$ being injective ($\phi$ is real valued and real Borel measurable and additional hypothesis may be imposed on it too). Specifically, I want to apply this formula to the case of finding the PDF of a function of a real valued random variable.-- Abdul Muhsy talk 04:01, 21 March 2022 (UTC)

OK I see that now.- Abdul Muhsy talk 08:59, 21 March 2022 (UTC)