Wikipedia:Reference desk/Archives/Mathematics/2010 February 6

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February 6

Complex integral with respect to conjugate

Hi all :)

I was wondering - I've worked out how to evaluate $\int _{|z|=r}f({\bar {z}})d{\bar {z}}$ for an analytic function f on $\mathbb {C}$ (I got the integral to be 0 by a few substitutions and Cauchy's theorem, if I did do it correctly) - but I was wondering whether it would be possible to evaluate $\int _{|z|=r}{\bar {f}}(z)d{\bar {z}}$ too - I can't quite get my head around it, because ${\bar {f}}$ wouldn't be analytic in the z plane, since f is, but then it would be presumably in the ${\bar {z}}$ plane which is where we're integrating - but then f is a function of z and not ${\bar {z}}$, so I'm not completely sure about what's going on and I think i'm overly confusing myself! I got to a point in my workings originally where I'd equated $\int _{|z|=r}{\bar {f}}(z)d{\bar {z}}$ to $-r^{2}\int _{|z|=r}{\frac {{\bar {f}}(z)}{(z-0)^{2}}}dz$ (if I did all my substitutions correctly!), and was about to use Cauchy's formula when i realised that ${\bar {f}}(z)$ isn't analytic so that doesn't behave according to Cauchy's formula, and now I'm back to square one unless I can salvage any of it!

Thanks very much for the help; 131.111.185.75 (talk) 01:13, 6 February 2010 (UTC)

I think since complex conjugation is an automorphism, $\int _{|z|=r}{\bar {f}}(z)d{\bar {z}}$ is equal to the complex conjugate of $\int _{|z|=r}f(z)dz.\,$ Rckrone (talk) 06:29, 6 February 2010 (UTC)

Substitute intermediately

${\bar {z}}=w$

to get

$\int _{|z|=r}f({\bar {z}})d{\bar {z}}=\int _{|w|=r}f(w)dw=\int _{|z|=r}f(z)dz$

$\int _{|z|=r}{\bar {f}}(z)d{\bar {z}}=\int _{|w|=r}{\bar {f}}({\bar {w}})dw={\int _{|z|=r}{\bar {f}}({\bar {z}})dz}$

Bo Jacoby (talk) 01:27, 7 February 2010 (UTC).

Yotoplex?

Hello I recall somewhere on wikipedia that there was a page which listed All the numerical number names ( Hundreds,Thousands,Millions And so on ) In size order, and at the bottem there was A Yotoplex. What is the Value of a Yotoplex? 94.172.41.229 (talk) 15:23, 6 February 2010 (UTC)

Here it lists "yotta" as an SI prefix for 10²⁴, so a yottametre for example would be a very big distance. That's the only "yot" on tht page so maybe it's what you're remembering?

Maybe you mean googolplex? AndrewWTaylor (talk) 22:37, 6 February 2010 (UTC)

It's possible you are remembering the article Names of large numbers, which at one time contained a bunch of number names that were just made up one day by somebody. It is possible that "yotoplex" was once listed there; the article's talk page contains a proposal for names like "yottillion" and "yottillio-illio-illion". But such number names seem to have been invented by the guy who proposed them, and they are not standard in any way. The problem of such made-up names appearing in the article was resolved by requiring number names listed in the article to be attributable to one or more mainstream dictionaries, so you won't find names like "yottillion" there today. —Bkell (talk) 05:30, 7 February 2010 (UTC)

I'm the inventor of the yottillio-illio-illion (a million raised to a million raised to the 10^24th power of a million) and I've never heard of a yotoplex.The latest development of my number-naming system is at my "Counting Really,Really,REALLY High" page which is listed among the "Web Projects" at put.com (trying to give the URL here would make it impossible to post this note,as the Captcha it would trigger is incompatible with my preferred browser Lynx).--Louis E./le@put.com/12.144.5.2 (talk) 19:49, 23 May 2010 (UTC)

modulo operation

Hi, In the article Modulo operation, it talks about the floor function and it gives:

$r=a-n(a/n)$

where r is the remainder, a the dividend and n the divisor. But I must be having a mind seize today because that looks like the n's will cancel and so r = a-a.

Can someone tell me what I'm misunderstanding? 74.227.108.180 (talk) 18:19, 6 February 2010 (UTC)

You left out the floor function in the expression. It's not $r=a-a(a/n)\,$, it's $r=a-n\left\lfloor {a \over n}\right\rfloor .$. The funny square brackets represent the floor function. --COVIZAPIBETEFOKY (talk) 18:35, 6 February 2010 (UTC)

Thanks, I didn't know the notation, I just thought it was like parens. I've got it now. 74.227.108.180 (talk) 18:45, 6 February 2010 (UTC)

Probability of each outcome when rolling 3 dice

I'm creating a campaign for a roleplaying game, and one of the levels involves gambling. I want the players to bet some amount of money on the outcome of 3 rolled dice. I want them to choose from one of 3 "blocks", or sets of outcomes. Originally, I had it 1-6, 7-12, 13-18, (i.e. if they bet 1-6, they roll the 3 dice and if the sum is between 1-6 they win), but I realized this isn't actually fair, since 7-12 has a much greater chance than 1-6 (you can't even roll a 1 or 2 with three dice). How should I divide the outcomes to get 3 blocks of equal or as-equal-as-possible outcomes? 169.231.8.128 (talk) 22:13, 6 February 2010 (UTC)

It is not possible to get exactly even thirds. If you include 3-8, there are 56 possible combinations. Similarly, there are 56 possible combinations for 13-18. 10-11 has 54 possible combinations. That leaves out 9 and 12, each with 25 possible combinations. If you stick 9 into the first or second group, you make one with a lot more possibilities than another one. -- kainaw™ 22:32, 6 February 2010 (UTC)

I could give you the formula for calculating probabilies in dice rolls, but this link might be easier. just add up the probabilities to get as close to 33% as possible. --Ludwigs2 22:33, 6 February 2010 (UTC)

I think what I'll do is make the blocks 3-8, 10-11, and 13-18. If they roll a 9 or a 12, they just lose. To compensate, I'll increase the winnings to 4-1 instead of 3-1. Does that work? It's the closest thing I can think of to even. 169.231.8.128 (talk) 22:55, 6 February 2010 (UTC)

If it works for you, it works. I'll note, however, that you don't need contiguous blocks - you could use any combination of numbers and call it a block. might be easier to balance probabilities that way. --Ludwigs2 23:20, 6 February 2010 (UTC)

Here's one non-contiguous way to divide it up into sixths. You can combine as you like for thirds or halves:

{3,6,9}; {4,5,10}; {7,8}; {13,14}; {11,16,17}; {12,15,18}.

And for quarters:

{5,8,10}; {3,4,6,7,9}; {12,14,15,17,18}; {11,13,16}.

Rckrone (talk) 00:19, 7 February 2010 (UTC)

One nice way to divide it up is by whether the sum is a multiple of 3, is 1 mod 3, or is 2 mod 3. Those each have a 1/3 probability. Similarly you can divide it up into sixths by the value mod 6. Rckrone (talk) 00:52, 7 February 2010 (UTC)

Is there any advantage to using 3 dice with 0,1,2 mod 3 as opposed to 1 die and choosing (say) 1,4 or 2,5 or 3,6? It surely isn't "more random". -- SGBailey - not signed in 82.45.16.156 (talk) 14:15, 7 February 2010 (UTC)