Wikipedia:Reference desk/Archives/Mathematics/2010 June 19

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June 19

Estimating a minimum from discrete points

Suppose I have a set of discrete values sampled from a function that has no slope discontinuities. I don't know the underlying function, all I have are data. The function looks roughly like a skewed hyperbolic curve with a minimum.

The minimum value sample I have isn't necessarily the minimum of the function. I would like to estimate the minimum (x,y) coordinate of the function somehow, without inventing a model underlying function to curve-fit the sampled values.

One way I can think to do this is to fit a cubic spline to the minimum 4 values and find the minimum of that. I was hoping to do it by projecting the slopes on either side of the minimum and seeing where they intersect, but this seems to work only when the sample values are symmetric around the minimum.

Is there some accepted way of doing this? ~Amatulić (talk) 07:06, 19 June 2010 (UTC)

I suppose you mean: Find (x,y) such that f(x,y) is minimum. But "without inventing a model underlying function to curve-fit the sampled values" makes the problem undefined. Bo Jacoby (talk) 11:28, 19 June 2010 (UTC).

By analogy with the German tank problem, you could try:

${\hat {Y}}=m-{\frac {m}{k}}+1$

to estimate the lowest population y value, where m is the lowest y value observed from k such values. Not that this would give the corresponding x value, nor may the underlying distribution of y values be appropriate - but it's easy.→81.147.3.245 (talk) 12:41, 19 June 2010 (UTC)

Fitting a cubic function to the points in the region of the minimum sounds reasonable to me. Can't see any reason for a cubic spline though if you're only interested in the minimum. It you only use four points you can fit a cubic exactly. If there's error (uncertainty) in the function values you might want to fit to more points by least squares. The choice of how many points is then a trade-off between the amount of error and the size of the region over which the function is well approximated by a cubic—some external info on the amount of measurement error would probably help. Qwfp (talk) 14:19, 19 June 2010 (UTC)

A cubic spline may be needed just to bias the slope of the cubic section that makes the minimum. I think my underlying function is well-behaved enough that I won't need it, though. ~Amatulić (talk) 19:11, 19 June 2010 (UTC)

Continuity of the inverse function

Let M, N ⊆ ℝⁿ open sets, and f: M → N a bijective continuous function. My question: Is the inverse function f^-1: N → M always continuous? --84.62.192.52 (talk) 16:02, 19 June 2010 (UTC)

Yes. This is essentially the theorem of invariance of domain. Algebraist 17:12, 19 June 2010 (UTC)

OK, you said open sets. The proposition is of course not true in more general situations. Michael Hardy (talk) 05:44, 20 June 2010 (UTC)

Equilateral planets

Hi, sorry if this question sounds dumb to you (or even homework-ey), but I'm not entirely sure how to do it myself (and it's definitely not homework). First of all, Planet Earth is ~27,000±1,000 ly away from the galactic center. Now, imagine there are other two planets which are exactly the same distance from the Galactic Center as Earth, and are also exactly apart from each other and Earth (i.e. they would make up the vertexes of an equilateral triangle). My question would be, what is the distance between each planet? I know there's an easy way to know this, but I couldn't come up with it... Thanks in advance! :) 186.80.207.31 (talk) 18:04, 19 June 2010 (UTC)

The distance between a planet and the center is the radius of the circumscribed circle of the triangle; the distance between each planet is its side. Therefore, the latter is ${\sqrt {3}}$ times the former. -- Meni Rosenfeld (talk) 18:26, 19 June 2010 (UTC)

Thank you, but could I ask why? I don't get how you got that number from what you said. 186.80.207.31 (talk) 19:05, 19 June 2010 (UTC)

In Equilateral triangle, it is said that when the side is a and the circumscribing radius is R you have $R={\frac {\sqrt {3}}{3}}a$ (this can be found with some trigonometry). Then $a={\frac {3}{\sqrt {3}}}R={\sqrt {3}}R$. -- Meni Rosenfeld (talk) 19:13, 19 June 2010 (UTC)

I understand now. Thanks again! 186.80.207.31 (talk) 19:58, 19 June 2010 (UTC)

Don't just take the article's word for it -- when MR says if can be found with some trigonometry, he means very simple trigonometry. See 30-60-90 triangle for a diagram showing how applying the Pythagorean theorem to a bisected equilateral triangle give you the length of the side opposite the 60° angle as √3/2 time that of the hypotenuse. (That is, sin 60° = √3/2.) Then make a diagram of your planets and form a triangle with one vertex at the galactic center and the other two at two of your planets. Note that the central angle is 1/3 of a circle, or 120°, and that if you bisect that angle you get a 30-60-90 triangle with a hypotenuse of length R = 27,000 ly and the side opposite the 60° angle of length half the distance between planets. But you know that this latter length is √3/2 R, so the distance between planets is √3 R. -- 58.147.53.127 (talk) 01:17, 20 June 2010 (UTC)

"when MR says if can be found with some trigonometry, he means very simple trigonometry", I wouldn't bet on that always being true though ;-) Dmcq (talk) 11:56, 20 June 2010 (UTC)

It should always be obvious to even the most dim-witted individual who holds an advanced degree in hyperbolic topology. :) -- Meni Rosenfeld (talk) 12:12, 20 June 2010 (UTC)

Is the integral of 1/x = ln(x)+C or ln|x|+C ?

What is the integral of ${\frac {1}{x}}$
$\ln x+C$
or
$\ln {|x|}+C$
? I get conflicting advice ––220.253.96.217 (talk) 22:42, 19 June 2010 (UTC)

Well, it depends on how you look at it. Indefinite integrals are a little bit hokey anyway; the things that make more sense are definite integrals and antiderivatives. If you're thinking of indefinite integrals as definite integrals for which the limits are not specified, then the version with the absolute-value signs work, provided the unspecified limits have the same sign. --Trovatore (talk) 22:45, 19 June 2010 (UTC)

Obviously it doesn't matter when x is real, but what if x is complex? That would cause ln(x) and ln|x| to have very different values.--220.253.96.217 (talk) 22:58, 19 June 2010 (UTC)

Oh, it's completely wrong for any complex-valued calculation. ("It" being the one with the absolute-value signs.) --Trovatore (talk) 23:02, 19 June 2010 (UTC)

So the correct integral is ln(x)+C then? Does that still work when x is negative? ––220.253.96.217 (talk) 23:11, 19 June 2010 (UTC)

In a complex-valued context, it "works", subject to the hassles regarding picking a branch of the log function. In a real-valued context it doesn't work, because the log of a negative number is undefined. --Trovatore (talk) 23:13, 19 June 2010 (UTC)

OK suppose I want to find the area underneath $y={\frac {1}{x}}$ between –2 and –4. That would be ln(–2) – ln(–4) = ln(2) + (2k-1)iπ - (ln(4) + (2k-1)iπ) = ln(2) – ln(4) = ln(1/2), this assumes you choose the same value for k when finding the ln of –2 and –4. Is this a fair assumption to make?––220.253.96.217 (talk) 23:33, 19 June 2010 (UTC) —Preceding unsigned comment added by 220.253.96.217 (talk) 23:32, 19 June 2010 (UTC)

Well, let me put it this way: Did you get the right answer? (In a picky sense, no, you didn't, because you got a negative number, and area can't be negative, but with luck a minute or two's reflection will sort that out for you — you did get the right answer for the definite integral.)

Now, was it just luck, or can you prove it happens in general? --Trovatore (talk) 08:23, 20 June 2010 (UTC)

For reals the |x| is correct but the C isn't the same on both sides of x=0 in that you can't integrate over that point. For complex numbers you can get from one side to the other by going one way or the other round the singularity and use the complex logarithm everywhere. Dmcq (talk) 23:25, 19 June 2010 (UTC)

It's log|x| + constant if x is real and not zero; and of course if x is positive then the absolute value sign can be dropped. Allowing x to be complex opens other cans of worms; as a function of complex x, the absolute value function is not differentiable anywhere.

And as someone noted above, the constant isn't "constant" on the whole line; it's one constant when x is positive and another when x is negative, and the two constants may or may not be equal. Michael Hardy (talk) 23:53, 19 June 2010 (UTC)

The correct answer is: For every simply connected domain D which doesn't contain the zero, the anti-derivative of the function $f={\frac {1}{x}}$ (where f is defined in D) is: Log(x)+C (where Log is any continuous function satisfying exp(Log(z))=z for every z in D), that is: ln|x|+iArg(x)+C (where ln is the real function satisfying exp(ln(x))=x for every positive x, and Arg is any continuous function satisfying tan(Arg(z))=Im(z)/Re(z) for every z in D which has a non-zero real part).

For example, the area underneath $f={\frac {1}{x}}$ (where f is defined in D) along a continuously differentiable path between 6 and -2 is:

(Log(6)+C)-(Log(-2)+C) = (ln(6)+ 2kiπ + C) - (ln(2) + (2k-1)iπ + C) = ln(6)-ln(2) + iπ = ln(3) + iπ.

HOOTmag (talk) 08:16, 20 June 2010 (UTC)

As I said above, the whole notion of an "indefinite integral" (that is, the things with the +C at the end) is a little hokey. It works if you know what it means and why, but if you apply it blindly you can get into trouble.

That's a bit what you're doing when you compute $\int _{-2}^{6}{\frac {dx}{x}}$ and come up with the value $\log 3+i\pi \,\!$. What exactly do you mean by that? It certainly isn't "the area under the curve"; obviously area cannot have a nonzero imaginary part. It isn't the Riemann integral or the Lebesgue integral of that function on that interval; neither of those is defined here (they do have a Cauchy principal value, but it isn't what you calculated).

Now, your answer, $\log 3+i\pi \,\!$, isn't completely wrong. It is the path integral of ${\frac {dz}{z}}$, along a path in the complex plane from −2+0i to 6+0i that travels below 0+0i and never wraps around it. I think that's right; I had to draw a picture in my head. If I drew it wrong, maybe change "below" to "above". --Trovatore (talk) 08:39, 20 June 2010 (UTC)

The quantity of area is a complex number if the area is defined underneath a curve of complex function. HOOTmag (talk) 10:34, 20 June 2010 (UTC)

I thought I had a good imagination about such things but that eludes me. Dmcq (talk) 16:13, 20 June 2010 (UTC)

LOL :) HOOTmag (talk) 16:30, 20 June 2010 (UTC)

What I hear you saying here is basically that you want to define the phrase "area under a curve" to mean the definite integral. I suppose you can do that if you want, though it isn't really standard, and not literally correct (whereas the "area under a curve", when the curve is a nonnegative real-valued continuous function, is really literally an area).

So then forget about areas, and look at what I wrote about the definite integral. Your computation of the definite integral is unambiguously wrong, unless you mean a path integral in the complex plane. But even then you have to take the right path to get the answer to come out the same as the one you got. --Trovatore (talk) 19:06, 20 June 2010 (UTC)

What do you mean by "the right path"? Is there a "wrong" path? Remember that we're talking about a function 1/x defined in a simply connected domain D which doesn't contain the zero, so what kind of "wrong" continuously differentiable path can there be inside D? HOOTmag (talk) 20:54, 20 June 2010 (UTC)

Say what? No we're not. When you take out zero, it's no longer simply connected. --Trovatore (talk) 20:56, 20 June 2010 (UTC)

Oh, wait, I think I see. You want to fix a simply-connected set to work in. Fine, if you pick the right one, then yes, you get the answer you got. But sorry, you can't call it "the area under the curve". No no no, that's just not going to fly at all. --Trovatore (talk) 21:00, 20 June 2010 (UTC)

What do you mean by "if you pick the right one"? Is there a "wrong" one - as long as a simply connected domain D (which doesn't contain the zero) is concerned? What I'm saying is that For every simply connected domain D which doesn't contain the zero, the area underneath $f={\frac {1}{x}}$ (where f is defined in D) along a continuously differentiable path between 6 and -2 is: ln(3) + iπ. Note that just as the quantity of area can be a real non-positive quantity (when the function is real and unnecessarily positive, e.g. check the two areas above and under the curve of f(x)=x between -3 and 2), so the quantity of area can be a complex non-real quantity (when the function is complex and unnecessarily real). HOOTmag (talk) 22:56, 20 June 2010 (UTC)

Then your statement is just wrong again, ignoring your curious phrase "area under the curve". The path integral in question can take any value of the form ln(3)+(2n+1)iπ (where n is any integer), depending on what domain D you choose (and thus what paths are available to integrate along). Algebraist 23:03, 20 June 2010 (UTC)

Look again: For every simply connected domain D which doesn't contain the zero, the anti-derivative of the function $f={\frac {1}{x}}$ (where f is defined in D) is: Log(x)+C (where Log is any continuous function satisfying exp(Log(z))=z for every z in D), that is: ln|x|+iArg(x)+C (where ln is the real function satisfying exp(ln(x))=x for every positive x, and Arg is any continuous function satisfying tan(Arg(z))=Im(z)/Re(z) for every z in D which has a non-zero real part).

For example, the area underneath $f={\frac {1}{x}}$ (where f is defined in D) along a continuously differentiable path between 6 and -2 is:

(Log(6)+C)-(Log(-2)+C) = (ln(6)+ 2kiπ + C) - (ln(2) + (2k-1)iπ + C) = ln(6)-ln(2) + iπ = ln(3) + iπ.

HOOTmag (talk) 23:08, 20 June 2010 (UTC)

Look again yourself. Suppose your domain D is a thin curved strip which goes n-and-a-half times anticlockwise around 0 on its way from -2 to 6. Then any curve from -2 to 6 in D will also go n-and-a-half times anticlockwise around 0, and so Arg will go up by (2n+1)iπ en route. Algebraist 23:11, 20 June 2010 (UTC)

Ah, nice. I hadn't been able to picture any way to get the full general-n case. I think you can also draw a wiggly branch cut that forces you to get this answer, but I can't quite nail it down in my mind's eye — I might actually have to pick up a pencil. --Trovatore (talk) 23:18, 20 June 2010 (UTC)

Then the area underneath $f={\frac {1}{x}}$ (where f is defined in D) along a continuously differentiable path between 6 and -2 is:

(ln(6)+ 2niπ + C) - (ln(2) + (2n+1)iπ + C) = ln(6)-ln(2) - iπ = ln(3) - iπ.

Note that whenever I wrote iπ I intended to include also the option of -iπ. Note that mathematics can't define the very distinction between i and -i, i.e. if you take every true/wrong mathematical statement, and replace i by -i and vice versa, then you get again a true/wrong mathematical statement (respectively).

HOOTmag (talk) 23:41, 20 June 2010 (UTC)

My answer was correct for the definition of "area under a curve" you gave above, which corresponded to the standard path integral, and I sketched a proof of this, which your comment here in no way refutes. I would ask what your comment here is supposed to do, but since you seem to have no interest in engaging with other people's mathematical arguments, and you also seem never to have studied elementary complex analysis, I will instead leave this conversation. Algebraist 23:52, 20 June 2010 (UTC)

My comment was about your claim that any curve from -2 to 6 in D would go n-and-a-half times anticlockwise around 0, and so Arg would go up by (2n+1)iπ en route. so I responded that in your case the area underneath $f={\frac {1}{x}}$ (where f is defined in D) along a continuously differentiable path between 6 and -2 would be (ln|6|+ 2niπ + C) - (ln|-2| + (2n+1)iπ + C) = ln(6)-ln(2) - iπ = ln(3) - iπ. HOOTmag (talk) 00:16, 21 June 2010 (UTC)

(ec) If you pick the right one, you get the answer you got. If you pick a different one, you might not. I think with this formulation there are two possible answers; namely, the one you got, $\log 3+i\pi \,\!$, which corresponds (I think) to a branch cut extending through the top half-plane, and $\log 3-i\pi \,\!$, which corresponds to putting the cut through the bottom half-plane.

As for area being possibly complex, don't be silly. Area is area. It cannot be negative, and it is always real. --Trovatore (talk) 23:07, 20 June 2010 (UTC)

When I wrote iπ I meant also the option of -iπ. Note that mathematics can't define the very distinction between i and -i, i.e. if you take every true/wrong mathematical statement, and replace i by -i and vice versa, then you get again a true/wrong mathematical statement (respectively).

Can't the area be negative? check the two areas above and under the curve of f(x)=x between -2 and 2. According to your attitude, the sum of these two areas would be a positive quantity, as opposed to what's accepted everywhere in Maths. HOOTmag (talk) 23:25, 20 June 2010 (UTC)

No, area cannot be negative. Don't be silly. A definite integral can be negative; area cannot.

It is true that there is a symmetry between i and −i, and your claim about mathematical statements and changing between them is true, given sufficient qualifications, which you haven't given. Nevertheless it is not possible for the same quantity to be both i and −i. --Trovatore (talk) 00:00, 21 June 2010 (UTC)

Check the two areas above and under the curve of f(x)=x between -2 and 2. Adopting your attitude, the sum of these two areas would be a positive quantity, as opposed to what's accepted everywhere in Maths.

Who said that i and -i are the same quantity? I just said that when I wrote iπ I intended to include also the option of -iπ. HOOTmag (talk) 00:16, 21 June 2010 (UTC)

As to your second paragraph, I'm starting to see a pattern here — you write false things, then claim later that you meant something else. Of course in this case, as Algebraist has explained to you, the thing you are now claiming is also false.

As to the first paragraph, yes, the sum of the two areas is positive, and this is accepted everywhere in mathematics; your claim to the contrary is simply wrong. The definite integral is zero, but the definite integral is not the area. --Trovatore (talk) 00:37, 21 June 2010 (UTC)

What "pattern"? which "false" things? I've never said that i is equal to -i, have I? I just said that when I wrote iπ I intended to include also the option of -iπ, because "mathematics can't define the very distinction between i and -i, i.e. if you take every true/wrong mathematical statement, and replace i by -i and vice versa, then you get again a true/wrong mathematical statement (respectively)". That's what I said. where do you see here "pattern" or "flase" things?

You assume that any area can't be negative, but it's not what's accepted in maths. For example, the area (not only the definite integral) of f(x)=x between -2 and 2 is considered in Maths to be zero.

HOOTmag (talk) 01:21, 21 June 2010 (UTC)

Did you also mean 3iπ, 5iπ, and 7iπ, or are you ignoring my point above? Algebraist 23:27, 20 June 2010 (UTC)

No, I'd referred to Trovatore's argument. As to your last argument, I have already referred to it just beyond it. Look above. HOOTmag (talk) 23:45, 20 June 2010 (UTC)

Could you explain, per my comment below, how your argument would go wrong for $x^{-2}$? — Carl (CBM · talk) 23:45, 20 June 2010 (UTC)

If you explain in a rigorous manner how you intend to apply my technique of $x^{-1}$ in $x^{-2}$ then I'll try to show the error. HOOTmag (talk) 00:16, 21 June 2010 (UTC)

Parallel to your argument: In any simply-connected domain excluding the origin, the antiderivative of $x^{-2}$ is $-x^{-1}$, so $\int _{-1}^{1}x^{-2}dx=(-1)-(1)=-2$. But, actually, that integral diverges. — Carl (CBM · talk) 00:28, 21 June 2010 (UTC)

Actually, Carl, I'm having a little trouble following this one too, given that $\int _{-1}^{1}x^{-1}dx$ also diverges. What distinction are you making here? --Trovatore (talk) 00:33, 21 June 2010 (UTC)

I was just picking a different integrand that avoids the logarithm issue, might be more well-known as a divergent integral, and is always positive on the real line minus the origin. The technique would work equally well for $x^{-3}$, etc. — Carl (CBM · talk) 00:41, 21 June 2010 (UTC)

Oh, now I've figured out what you mean. So, my technique is intended to deal with the function f(x)=1/x areas defined by path integrals only (the path being continuously differentiable). HOOTmag (talk) 01:21, 21 June 2010 (UTC)

You have not figured out what I mean. What I mean is that the definite integral you are talking about diverges. — Carl (CBM · talk) 01:42, 21 June 2010 (UTC)

I have figured out. I'm not talking about definite integrals at all, but rather about areas defined by path integrals (the path being continuously differentiable). HOOTmag (talk) 02:04, 21 June 2010 (UTC)

But as Algebraist indicated, the value of this path integral depends on the exact shape of the path, in particular on how many times it winds around the origin. It does not depend only on the endpoints. — Carl (CBM · talk) 02:33, 21 June 2010 (UTC)

Algebraist claimed that any curve from -2 to 6 in D would go n-and-a-half times anticlockwise around 0, and so Arg would go up by (2n+1)iπ en route. so I responded him that in his case the area underneath $f={\frac {1}{x}}$ (where f is defined in D) along a continuously differentiable path between 6 and -2 would be (ln|6|+ 2niπ + C) - (ln|-2| + (2n+1)iπ + C) = ln(6)-ln(2) - iπ = ln(3) - iπ. HOOTmag (talk) 20:13, 22 June 2010 (UTC)

And you're wrong. --Trovatore (talk) 20:54, 22 June 2010 (UTC)

The fact that I'm wrong is equivalent to the fact that you are the smartest person in the world. HOOTmag (talk) 21:32, 22 June 2010 (UTC)

That's also wrong. Does my saying you're wrong qualify me as well? As to specifics the difference would depend on the layer you take for each side so in your terms it is more like (ln|6|+ 2niπ + C) - (ln|-2| + (2m+1)iπ + C) = ln 6 - ln 2 +2(n-m)-1)iπ Dmcq (talk) 21:47, 22 June 2010 (UTC)

He's not the smartest one, so there's no contradiction in the equivalence.

Anyways, adopting your terms, I claim that n=m. Had it not been the case, the path-integral-function (in the simply connected domain D which doesn't contain the origin) wouldn't have been continuous, and consequently it wouldn't have been a path-integral-function (in D) at all.

HOOTmag (talk) 22:09, 22 June 2010 (UTC)

If you're thinking of a branch cut, they don't have to be straight. It can form a spiral out from the origin for instance. Dmcq (talk) 22:29, 22 June 2010 (UTC)

I'm talking about a "regular" (i.e. bi-dimensional) path integral (the path being continuously differentiable). Spirals need more than two dimensions. HOOTmag (talk) 23:06, 22 June 2010 (UTC)

You just keep digging.... --Trovatore (talk) 00:58, 23 June 2010 (UTC)

digging or not, my point (which hasn't been refuted yet) is that for every simply connected domain D which doesn't contain the zero, the anti-derivative of the function $f={\frac {1}{x}}$ (where f is defined in D) is: Log(x)+C (where Log is any continuous function satisfying exp(Log(z))=z for every z in D), that is: ln|x|+iArg(x)+C (where ln is the real function satisfying exp(ln(x))=x for every positive x, and Arg is any continuous function satisfying tan(Arg(z))=Im(z)/Re(z) for every z in D which has a non-zero real part).

My second point is that the "regular" (i.e. bi-dimensional) path integral of $f={\frac {1}{x}}$ (where f is defined in D the path is continuously differentiable) between 6 and -2, can be computed - using the familiar technique of anti-derivatives, this way:

$\int _{-2}^{6}{\frac {dx}{x}}$ = (Log(6)+C)-(Log(-2)+C) = (ln(6)+ 2niπ + C) - (ln(2) + (2n ± 1)iπ + C) = ln(6)-ln(2) ± iπ = ln(3) ± iπ.

HOOTmag (talk) 07:26, 23 June 2010 (UTC)

You should look at the articles Cauchy's integral formula and Residue theorem-- these are topics in complex analysis you seem to be missing. The function f(z) = 1/z has a pole at z=0 (since the denominator is z) with a residue of 1 (since the numerator is 1). This means that any path integral of f(z) that goes counterclockwise around z=0 once has the value 2πi.

Since line integrals are additive over paths, if the path goes counterclockwise around z=0 n times, the value is 2nπi, if it goes clockwise around z=0 n times, the value is -2nπi. (This n is the Winding number.) When you choose a path between z=6 and z=-2, as you know, you can't go straight across the origin because of the singularity at z=0. You can go above or beneath z=0, and it changes the final answer: ln(3)+πi or ln(3)-πi. You can loop around z=0 as many times as you want, which allows us to get ln(3)+(2n+1)πi for any integer n. The answers above that you reject really are correct, and it's important to understand why, or you'll miss something basic about the line integral you propose.

With line integrals, you should expect that a different path can give you a different answer. You need Cauchy's integral theorem to show that in this case, all we need to know about the path is how many times it loops around z=0. You should work through the proof of this theorem and the Residue Theorem to really understand your question; I'm sure that those who responded above have personally worked through the proofs of these theorems at some point. 98.235.80.144 (talk) 15:34, 23 June 2010 (UTC)

It looks like you haven't read all of what I've written, because I did refer to all of that many times, and once again my readers ignore the fact that I'm talking about a function which is defined in a simply connected domain which doesn't contain the zero, the path being inside this domain. So please read now our article Cauchy's integral theorem, and see that for every function f which is holomorphic in a simply connected domain D, F being f's anti-derivative, the integral of f between a and b, along a continuously differentiable path γ (in D), is: $\int _{\gamma }f(z)\,dz=F(b)-F(a).$. Conclusion: the path integral is path independent. HOOTmag (talk) 18:58, 23 June 2010 (UTC)

No-one is ignoring this fact. Everyone agrees that if you fix a domain D (simply connected, does not contain 0), then the value of the integral is independent of the choice of the path in D. But if you choose a different domain E (again simply connected and not containing 0), the value of the integral for paths in E may be different from the value obtained from paths in D. Example: if you choose a "fat semicircle" above the real axis the itegral is log(3)+iπ, and if you reflect this domain in the real axis the integral is log(3)-iπ. Tinfoilcat (talk) 19:14, 23 June 2010 (UTC)

You have ignored my computation, which coincide with yours. So let me quote myself:

$\int _{-2}^{6}{\frac {dx}{x}}$ = (Log(6)+C)-(Log(-2)+C) = (ln(6)+ 2niπ + C) - (ln(2) + (2n ± 1)iπ + C) = ln(6)-ln(2) ± iπ = ln(3) ± iπ.

How many times should I have to repeat this "±" ?

HOOTmag (talk) 19:27, 23 June 2010 (UTC)

I have not ignored your computation. Firstly you contradict yourself by saying 1) "the path integral is path independent", and 2) there are two different possibilities for the path integral (the ±). Secondly, there are infinitely many other possible values of the integral, depending on different choices of domain. Let me add another example: take a path that starts at 6, travels below the real axis then crosses it between -2 and 0, curves back again above the real axis so that it meets it to the right of 6, then proceeds below the real axis to -2. Then the value of the integral is...? Tinfoilcat (talk) 19:50, 23 June 2010 (UTC)

(ec) No, the different possible values have nothing to do with the fact that "i" and "-i" are algebraically indsitinguishable over the real numbers (i.e., the fact that complex conjugation is a ring automorphism of the complex numbers that fixes the real numbers).

Choosing a domain D and choosing a branch cut for the complex logarithm are different ways of saying the same idea, and it seems to us that you're missing this idea.

Below, I have drawn a path from -2 to 6 in the complex plane. This path is a simply connected domain D, not containing 0; and on this path, the integral is ln(3)-3πi.

Notice that $e^{-3\pi i}=-1$, so -3πi is one of the possible values for ln(-1). But you need to specify the branch cut or the domain D to know which value to use. On the domain D below, the only correct answer is ln(3)-3πi. It is not ln(3)+3πi or ln(3) ± πi.

                                             _____|_________             
                                            /     |         \          
                                           /   ___|__        \         
                                          /   /   |  \        \        
                                   ------(---|----|---)--------|-------
                                          \ -2    |  /         6       
                                           \______|_/                  
                                                  |                    

If you want to see the details of this example explicitly, the path I've drawn can be parametrized by $z={\frac {2+4t}{3}}e^{-\pi it}$ for t from 1 to 4.

Then the differential dz is $dz={\frac {4-\pi i(2+4t)}{3}}e^{-\pi it}dt$.

So we have $\int _{-2}^{6}{\frac {dz}{z}}=\int _{1}^{4}{\frac {4-\pi i(2+4t)}{2+4t}}dt=\int _{1}^{4}{\frac {4}{2+4t}}dt+\int _{1}^{4}-\pi idt=\ln(3)-3\pi i$.

98.235.80.144 (talk) 20:25, 23 June 2010 (UTC)

HOOTmag: Let us consider the following simply connected domain D which does not contain the origin: It contains all points except those on the spiral $\{\theta \exp(100i\theta )|\theta \geq 0\}$. Any path from -2 to 6 in this domain circles many times around the origin, and the integral along such path will have a large coefficient for the imaginary part.

Your mistake is that you think $\arg(z)$ must be between $-\pi i$ and $\pi i$ (or likewise). In fact, for a domain like the one I described, it takes a wide range of values. -- Meni Rosenfeld (talk) 09:17, 24 June 2010 (UTC)

No, this is not my mistake, because in my first post here I made it clear that the anti-derivative of the function $f={\frac {1}{x}}$ is: ln|x|+iArg(x)+C, and then I claimed that the area underneath $f={\frac {1}{x}}$ along a continuously differentiable path between 6 and -2 is: (ln(6)+ 2kiπ + C) - (ln(2) + (2k±1)iπ + C). So how could I have thought that $\arg(z)$ must be between $-\pi i$ and $\pi i$ (or likewise), after my clear words about the constant k involved in describing the function Arg?

If you really want to know about my thoughts, look at my response on 98.235.80.144's talk page.

HOOTmag (talk) 09:53, 24 June 2010 (UTC)

That was the "likewise" part - you thought that arg was always restricted to a range of $2\pi$, or something similar.

It looks like you understand now. People have tried to explain to you that there are domains like this as early as the post by Algebraist 23:11, 20 June, it's unfortunate that you have resisted for this long. -- Meni Rosenfeld (talk) 10:11, 24 June 2010 (UTC)

A well known proverb, in an ancient language, goes like this: LO HA-BIE-SHAHN LA-MED...

I resist as long as I feel that other people's arguments don't relate to my own argument. e.g. Algebraist's (and others') argument about the spiral was not intended to refute my thought about the constant k, but rather to refute other opinions which were mistakenly ascribed to me, while 98.235.80.144 did refer to my position.

HOOTmag (talk) 10:32, 24 June 2010 (UTC)

Just as a heuristic: any solution for $\int _{-1}^{1}x^{-1}dx$ that would work in the same way and give a finite value for $\int _{-1}^{1}x^{-2}dx$ cannot possibly be correct, because the latter integral diverges. But HOOTmag's technique would appear to work inthe same way for both of these integrals. The problem is that these integrals correspond to a path along the x axis, right through the singularity. — Carl (CBM · talk) 23:15, 20 June 2010 (UTC)