Math Challenge - January 2020

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  • #51
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In general, any ##2\pi## periodic function can be translated to be zero at the end points, with the same integral and derivative. The result holds in general, therefore, with equality iff:
Nitpick: Not sure if I agree with this. Translating ##f(x)\to f(x)-c## does change the integral ##\int_0^{2\pi} |f(x)|^2 dx##, and anyway ##\int_0^{2\pi}f(x) dx## wouldn't be zero anymore. Did you have something else in mind?

Edit: Nevermind, I assume you mean a translation ##f(x)\to f(x-c)##. Your solution is fine :)
 
  • #52
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Nitpick: Not sure if I agree with this. Translating ##f(x)\to f(x)-c## does change the integral ##\int_0^{2\pi} |f(x)|^2 dx##. Did you have something else in mind?
##f(x) \rightarrow f(x -c)##
 
  • #53
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From the conditions on ##f_k(x)## that are given, which are the convexity, ##f_k(0)=0##, ##f_k(x)\geq 0##, we have that, for all ##k##, ##f_k(x)## are non-decreasing functions
Now from the definition of convexity, we have:
$$f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)f(y)$$
for all ##x## and ##y## in the domain of ##f##, and ##\lambda \in [0,1]##.
We prove that a product of two convex non-decreasing positive functions, is also convex(we take that they are defined on the same domain).
Proof:
Let's take ##f## and ##g## to be positive, non-decreasing, convex functions. Then since they are non-decreasing, we have for any two points in their domain:
$$(f(x)-f(y))(g(x)-g(y)) \geq 0 \Leftrightarrow f(x)g(x) + f(y)g(y) \geq f(x)g(y) + f(y)g(x)$$
Now we apply the definition of convexity to check if their product is also convex, using the above inequality:
$$f(\lambda x + (1-\lambda)y)g(\lambda x + (1-\lambda)y) \leq (\lambda f(x) + (1-\lambda)f(y))(\lambda g(x) + (1-\lambda)g(y)) = \lambda^2f(x)g(x) + \lambda(1-\lambda)(f(x)g(y) + f(y)g(x)) + (1-\lambda)^2 f(y)g(y) \\ \leq \lambda f(x)g(x) + (1-\lambda)f(y)g(y)$$

Now that we have proven that the product of two functions from our set of functions in the problem is also convex(and obviously trivially satisfying all the other conditions of the set ##M##), we have that it is sufficient to prove the inequality for two functions that belong to ##M##. The extension to any finite number of them then follows by induction.
We now construct the 'hat' function by:
$$\hat{f}(x) = 2x\int_0^1 f(x)dx$$
and we observe that: ##\int_0^1 \hat{f}(x)dx = \int_0^1 f(x)dx##.
Then our inequality boils down to proving that for two functions ##f## and ##g## belonging to ##M##, we have:
$$\int_0^1 \hat{f}(x)\hat{g}(x)dx \leq \int_0^1 f(x)g(x)dx$$

Proof:
We observe that, because ##f \in M## is convex and non-decreasing and positive, and ##\hat{f}## is linear and has the same integral over ##[0,1]## as ##f##, we can't have ##f(x) > \hat{f}(x)## on the whole domain, nor can we have ##f(x) < \hat{f}(x)## on the whole domain. That is, neither of the two functions can dominate the other on the whole domain. There must be a point where they are equal, or if this change happens at a jump discontinuity, then there must exist this point, ##x=a##, that divides the domain into two parts(from the monotoneity of ##f##) such that:
$$f(x) < \hat{f}(x) , x<a \qquad f(x)>\hat{f}(x) , x>a$$
Then, it is easy to derive the following inequality:
$$\int_0^1 [f(x)-\hat{f}(x)]g(x)dx = \int_0^a [f(x)-\hat{f}(x)]g(x)dx + \int_a^1 [f(x)-\hat{f}(x)]g(x)dx \geq g(a)\int_0^a [f(x)-\hat{f}(x)]dx + g(a)\int_a^1 [f(x)-\hat{f}(x)]dx = 0$$
Where the inequality follows from non-decreasing property of ##g## and the inequality we derived above, and the final equality is consequence of the equality of integrals of ##f## and ##\hat{f}##.
From this, we conclude:
$$\int_0^1 f(x)g(x)dx \geq \int_0^1 \hat{f}(x)g(x)dx \geq \int_0^1 \hat{f}(x)\hat{g}(x)dx$$
In the last inequality, we proceeded analogous to the derivation above, using that ##\hat{f}## is increasing for any ##f \in M##.
By induction we obtain iteratively for ##n## functions that belong to ##M##:
$$\int_0^1 \prod_{k=1}^n f_k(x)dx \geq \int_0^1 \prod_{k=1}^n \hat{f_k}(x)dx = \frac{2^n}{n+1} \prod_{k=1}^n \int_0^1 f_k(x)dx$$
which is the result we needed.
 
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  • #54
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From the conditions on ##f_k(x)## that are given, which are the convexity, ##f_k(0)=0##, ##f_k(x)\geq 0##, we have that, for all ##k##, ##f_k(x)## are non-decreasing functions
Yes, this is correct. Can you add, why the functions in ##M## are not decreasing anywhere?
Now from the definition of convexity, we have:
$$f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)f(y)$$
for all ##x## and ##y## in the domain of ##f##, and ##\lambda \in (0,1)##.
##\lambda \in [0,1]##
We prove that a product of two convex non-decreasing positive functions, is also convex(we take that they are defined on the same domain).
Proof:
Let's take ##f## and ##g## to be positive, non-decreasing, convex functions. Then since they are non-decreasing, we have for any two points in their domain:
$$(f(x)-f(y))(g(x)-g(y)) \geq 0 \Leftrightarrow f(x)g(x) + f(y)g(y) \geq f(x)g(y) + f(y)g(x)$$
Now we apply the definition of convexity to check if their product is also convex, using the above inequality:
$$f(\lambda x + (1-\lambda)y)g(\lambda x + (1-\lambda)y) \leq (\lambda f(x) + (1-\lambda)f(y))(\lambda g(x) + (1-\lambda)g(y)) = \lambda^2f(x)g(x) + \lambda(1-\lambda)(f(x)g(y) + f(y)g(x)) + (1-\lambda)^2 f(y)g(y) \\ \leq \lambda f(x)g(x) + (1-\lambda)f(y)g(y)$$

Now that we have proven that the product of two functions from our set of functions in the problem is also convex(and obviously trivially satisfying all the other conditions of the set ##M##), we have that it is sufficient to prove the inequality for two functions that belong to ##M##. The extension to any finite number of them then follows by induction.
We now construct the 'hat' function by:
$$\hat{f}(x) = 2x\int_0^1 f(x)dx$$
and we observe that: ##\int_0^1 \hat{f}(x)dx = \int_0^1 f(x)dx##.
Then our inequality boils down to proving that for two functions ##f## and ##g## belonging to ##M##, we have:
$$\int_0^1 \hat{f}(x)\hat{g}(x)dx \leq \int_0^1 f(x)g(x)dx$$

Proof:
We observe that, because ##f \in M## is convex and non-decreasing and positive, and ##\hat{f}## is linear and has the same integral over ##[0,1]## as ##f##, we can't have ##f(x) > \hat{f}(x)## on the whole domain, nor can we have ##f(x) < \hat{f}(x)## on the whole domain. That is, neither of the two functions can dominate the other on the whole domain. There must be a point where they are equal, or if this change happens at a jump discontinuity, then there must exist this point, ##x=a##, that divides the domain into two parts(from the monotoneity of ##f##) such that:
$$f(x) < \hat{f}(x) , x<a \qquad f(x)>\hat{f}(x) , x>a$$
1578768988032.png

Then, it is easy to derive the following inequality:
$$\int_0^1 [f(x)-\hat{f}(x)]g(x)dx = \int_0^a [f(x)-\hat{f}(x)]g(x)dx + \int_a^1 [f(x)-\hat{f}(x)]g(x)dx \geq g(a)\int_0^a [f(x)-\hat{f}(x)]dx + g(a)\int_a^1 [f(x)-\hat{f}(x)]dx = 0$$
Where the inequality follows from non-decreasing property of ##g## and the inequality we derived above, and the final equality is consequence of the equality of integrals of ##f## and ##\hat{f}##.
From this, we conclude:
$$\int_0^1 f(x)g(x)dx \geq \int_0^1 \hat{f}(x)g(x)dx \geq \int_0^1 \hat{f}(x)\hat{g}(x)dx$$
In the last inequality, we proceeded analogous to the derivation above, using that ##\hat{f}## is increasing for any ##f \in M##.
By induction we obtain iteratively for ##n## functions that belong to ##M##:
$$\int_0^1 \prod_{k=1}^n f_k(x)dx \geq \int_0^1 \prod_{k=1}^n \hat{f_k}(x)dx = \frac{2^n}{n+1} \prod_{k=1}^n \int_0^1 f_k(x)dx$$
which is the result we needed.
 
  • #55
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Thanks for the remark, I corrected ##\lambda## in an edit(it was typo).
As for the proof why ##f \in M## are non-decreasing, I will add it here.

Let's take ##f : [0,1] \rightarrow \mathbb{R}##, such that ##f## is convex, nonnegative and ##f(0) = 0##. Assuming that ##f## is not constant and equal to zero, which is a trivial case, ##f## must be non-decreasing at one part of the interval ##[0,1]##(say from ##x=0## to ##x=m##), since it is nonnegative and is beginning from zero value. Assume that somewhere in the domain there is a local maximum(which means that this non-decreasing trend turned into a decreasing one). Then there are points ##x=a<m## and ##x=b>m## in some neighbourhood of ##m## such that ##f(a)<f(m)## and ##f(b)<f(m)##(this follows from the definition of a local maximum).
Then we apply the convexity condition to ##a## and ##b## and we take ##\lambda## such that:
$$m = \lambda a + (1-\lambda)b \Rightarrow \lambda = \frac{b-m}{b-a}$$
Then:
$$f(m) = f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda)f(b) = \frac{b-m}{b-a}f(a) + \frac{m-a}{b-a}f(b)$$
But then from ##f(m) > f(b)## we have:
$$f(b) < \frac{b-m}{b-a}f(a) + \frac{m-a}{b-a}f(b) \Leftrightarrow f(b)<f(a)$$
and also from ##f(m)>f(a)## we have:
$$f(a) < \frac{b-m}{b-a}f(a) + \frac{m-a}{b-a}f(b) \Leftrightarrow f(a)<f(b)$$
a contradiction.
Therefore, ##f## has no local maxima, and hence it is non-decreasing on the whole domain.
 
  • #56
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62
Since no restrictions on the function ##f## are given, except that it is periodic with period ##\pi##, I will assume(reasonably, I hope), that we can expand the function into convergent Fourier series(except at a few points, possibly). This means that ##f## is assumed to be of bounded variation(if I remember correctly). We will expand ##f## over the interval ##[-\frac{\pi}{2}, \frac{\pi}{2}]##.
$$f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n cos(2nx) + \sum_{n=1}^\infty b_n sin(2nx)$$
Substituting this series, we can integrate it term by term.
Integral of the sine series is equal to zero, since it is an odd function integrated over a symmetric interval:
$$\int_{-\infty}^{\infty} sin(2nx) \frac{sin(x)}{x}dx = 0$$
Integral of cosine series is proven to be zero in the following way:
$$\int_{-\infty}^{\infty} cos(2nx) \frac{sin(x)}{x}dx = \int_{-\infty}^{\infty} \frac{sin((2n+1)x) - sin((2n-1)x)}{x}dx = 0$$
where the last equality is obtained with a simple substitution ##t = (2n \pm 1)x## using the known value of the sine integral ##\int_{-\infty}^{\infty} \frac{sin(x)}{x}dx = \pi##.
Therefore, the only term left in the series is the constant term, so we have:
$$a_0 = \frac{2}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(x)dx$$
$$\int_{-\infty}^{\infty} f(x)\frac{sin(x)}{x}dx = \frac{a_0\pi}{2} = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(x)dx$$
Finally we use that for periodic function with period ##T## we have ##\int_0^T f(x)dx = \int_a^{a+T} f(x)dx##, so we complete our proof by translating the boundary:
$$\int_{-\infty}^{\infty} f(x)\frac{sin(x)}{x}dx = \int_0^\pi f(x)dx$$

As for the second identity, I am suspicious that it won't work unless we put some restrictions on the function ##f##. Take ##f(x) = cos(2x)##, which is a function with period ##\pi##.
Then we have that the function in the integrand ##cos(2x)\frac{tan(x)}{x}## is not bounded at infinity, because we can always get tangent function to dominate the ##\frac{cos(2x)}{x}## function for arbitrarily large ##x##. So our integral doesn't converge.
However, ##\int_0^\pi cos(2x)dx = 0##, so the identity can't hold for this function.
 
  • #57
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Edit: Was unable to edit since I forgot about it, but in the cosine-series part there is one half missing that is inconsequential to the proof.
 
  • #58
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Since no restrictions on the function ##f## are given, except that it is periodic with period ##\pi##, I will assume(reasonably, I hope), that we can expand the function into convergent Fourier series(except at a few points, possibly). This means that ##f## is assumed to be of bounded variation(if I remember correctly). We will expand ##f## over the interval ##[-\frac{\pi}{2}, \frac{\pi}{2}]##.
$$f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n cos(2nx) + \sum_{n=1}^\infty b_n sin(2nx)$$
Substituting this series, we can integrate it term by term.
Integral of the sine series is equal to zero, since it is an odd function integrated over a symmetric interval:
$$\int_{-\infty}^{\infty} sin(2nx) \frac{sin(x)}{x}dx = 0$$
Integral of cosine series is proven to be zero in the following way:
$$\int_{-\infty}^{\infty} cos(2nx) \frac{sin(x)}{x}dx = \int_{-\infty}^{\infty} \frac{sin((2n+1)x) - sin((2n-1)x)}{x}dx = 0$$
where the last equality is obtained with a simple substitution ##t = (2n \pm 1)x## using the known value of the sine integral ##\int_{-\infty}^{\infty} \frac{sin(x)}{x}dx = \pi##.
Therefore, the only term left in the series is the constant term, so we have:
$$a_0 = \frac{2}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(x)dx$$
$$\int_{-\infty}^{\infty} f(x)\frac{sin(x)}{x}dx = \frac{a_0\pi}{2} = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(x)dx$$
Finally we use that for periodic function with period ##T## we have ##\int_0^T f(x)dx = \int_a^{a+T} f(x)dx##, so we complete our proof by translating the boundary:
$$\int_{-\infty}^{\infty} f(x)\frac{sin(x)}{x}dx = \int_0^\pi f(x)dx$$

As for the second identity, I am suspicious that it won't work unless we put some restrictions on the function ##f##. Take ##f(x) = cos(2x)##, which is a function with period ##\pi##.
Then we have that the function in the integrand ##cos(2x)\frac{tan(x)}{x}## is not bounded at infinity, because we can always get tangent function to dominate the ##\frac{cos(2x)}{x}## function for arbitrarily large ##x##. So our integral doesn't converge.
However, ##\int_0^\pi cos(2x)dx = 0##, so the identity can't hold for this function.
Yes, the second identity is problematic. The first one is called Lobachevski's formula which can be generalized to even powers of sine. I found the second one at the same place as the first one. However, it needs Fubini and a suitable pole handling, which I was hoping to summarize with the additional condition "assuming the integrals exist".

Means: If we only use integrals and series and ignore any condition on exchangeability of integral, series and limits, or finiteness, and integrate from pole to pole, then the identity is true. So this will give us a warning how easily mistakes can be made, if such conditions are not checked. Same as I did not check the details. I'd rather made a "find the mistake" problem out of it.
 
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  • #59
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62
Okay, well, I wasn't sure, since the integral was divergent on a few examples, but if we take that the function is very suitably defined as you said, and that we're looking for the Cauchy principal value of the integral by integrating from pole to pole, I will add such a proof below, just for the sake of completeness and curiosity of someone who would want to read it in the future. The proof is not extremely rigorous, since we're allowing the function to behave properly, evading complications as it was intended in the exercise. Also the proof is missing the picture of the contour, but one should have no problem drawing it with the directions given.

Regards!

We assume as below, that we have convergent Fourier series for ##f##, so that it can be integrated term by term, and we assume that the integrals in the exercise are convergent, that is, that on the left hand side we have Cauchy principal value of the integral.
Then:
$$f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos(2nx) + \sum_{n=1}^\infty b_n \sin(2nx)$$
From the convergence of the integral, we conclude that the sine series will give zero integrating term by term, since it is an odd function on a symmetric interval.
For the cosine part, the term we're integrating looks like:
$$\int_{-\infty}^{\infty} \cos(2nx) \frac{\tan(x)}{x}dx $$
Now we will search for principal value by using contour integration. Our contour will be anti-clockwise directed semicircle in the upper-half plane with infinite diameter along the real axis of the complex plane(a very standard contour, I don't have any good way to draw it here, so I assume the description of it is sufficient).
The function we will be integrating along this contour will be:
$$f(z) = e^{i(2nx)}\frac{\sin(z)}{z\cos(z)}$$
This function has simple poles at ##z = 0## and ##z = k\pi + \tfrac{\pi}{2}##, all along the real axis, so we will evade all of these poles by constructing small semi-circles in the upper-half plane around each pole, clockwise. This way, we won't have any poles inside the contour.
So, denoting the small semi-circles by ##C_0## and ##C_k##, the real axis part of the contour by ##\gamma##, the big semi-circle by ##C_R##, and the full contour by ##C##, we have:
$$\oint_C f(z)dz = \oint_{C_0} f(z)dz + \sum_{k=-m+1}^{m} \oint_{C_k}f(z)dz + \oint_{\gamma} f(z)dz + \oint_{C^+} f(z)dz$$
By Cauchy's residue theorem the total integral is zero since we have no poles inside the contour.
For the semicircles we will use Jordan's lemma. For now, we will assume the contour to be of finite width ##2R## with ##2m## poles in between(##k = -(m-1), \dots, m##), and in the end we will extend this contour to infinity.
For small semicircles, we have:
$$\lim_{r_k \rightarrow 0}\oint_{C_k} f(z)dz = -i\pi Res(f,k\pi+\tfrac{\pi}{2})$$
$$\lim_{r \rightarrow 0}\oint_{C_0} f(z)dz = -i\pi Res(f,0)$$
where we denoted the radii of the semicircles around the poles with ##r_k## and ##r##.

We calculate the residues:
$$Res(f,0) = \lim_{z \rightarrow 0} zf(z) = 0$$
$$Res(f,k\pi+\tfrac{\pi}{2}) = \lim_{z \rightarrow k\pi + \tfrac{\pi}{2}} \left(z - k\pi -\frac{\pi}{2}\right)f(z) = e^{i(2n(k\pi+\tfrac{\pi}{2}))}\sin\left(k\pi + \frac{\pi}{2}\right)\frac{2}{(2k+1)\pi} \lim_{z \rightarrow k\pi + \tfrac{\pi}{2}} \frac{z - k\pi - \frac{\pi}{2}}{\cos(z)} = (-1)^{(n+k)}\frac{2}{(2k+1)\pi} (-1)^{(k+1)} = \frac{2(-1)^{(n+1)}}{(2k+1)\pi}$$
Integral along the real axis(which we denoted by ##\gamma##) is:
$$\oint_{\gamma} f(z)dz = \int_{-R}^{R}e^{i(2nx)}\frac{\sin(x)}{x\cos(x)}dx$$
Combining the results above, we have:
$$ 0 = \oint_{C_R}f(z)dz + \int_{-R}^{R}e^{i(2nx)}\frac{\sin(x)}{x\cos(x)}dx +(-1)^n\sum_{k=-m+1}^{m} \frac{2i}{2k+1}$$
The integral we're looking for is the real part of the principal value we see in the expression above with boundaries going to infinity. When we take boundaries to infinity, ##C_R## integral will drop to zero, because by closing the contour anticlockwise in the upper-half of the plane, we obtained exponential damping of the integrand, so this limit will follow from Jordan's lemma. The sum that we have in the last term will diverge, however this sum is purely imaginary, the real part will still be zero. So we conclude that the real part of our integral is zero:
$$p.v. \int_{-\infty}^{\infty} \cos(2nx)\frac{\sin(x)}{x\cos(x)}dx = 0$$
This means that our cosine series will be integrated to zero term by term. We're left only with the constant term. So we have:
$$\int_{-\infty}^{\infty} f(x)\frac{\tan(x)}{x}dx = \frac{a_0}{2}\int_{\infty}^\infty \frac{\tan(x)}{x}dx$$
Where we're looking for the Cauchy principal value of the integral, as before. We will perform the same type of calculation as with the cosine series, this time working with the function:
$$f(z) = \frac{e^{iz}}{z\cos(z)}$$
The contour and method are completely identical, so we will proceed to calculate the residues:
$$Res(f,0) = 1$$
$$Res(f,k\pi + \frac{\pi}{2}) = \frac{-2i}{(2k+1)\pi}$$
From Cauchy integral theorem, we find:
$$0 = \oint_{C_R}f(z)dz -i\pi -\sum_{k=-m+1}^m \frac{2}{(2k+1)} + \int_{R}^{R} \frac{e^{ix}}{x\cos(x)}dx$$
Letting ##R## go to infinity, we find that the integral along the big semicircle ##C_R## goes to zero, analogous to the cosine-series case, and that the sum is divergent. However, the integral we're computing is imaginary part of the principal value in the formula above, and the sum is purely real. So we conclude:
$$p.v. \int_{-\infty}^{\infty} \frac{\tan(x)}{x}dx = \pi$$
Finally, we obtain:
$$p.v. \int_{-\infty}^{\infty} f(x)\frac{\tan(x)}{x}dx = \frac{\pi a_0}{2} = \int_0^\pi f(x)dx$$
So the correct identity is with Cauchy principal value on the left.
 
  • #60
12,533
8,948
Okay, well, I wasn't sure, since the integral was divergent on a few examples, but if we take that the function is very suitably defined as you said, and that we're looking for the Cauchy principal value of the integral by integrating from pole to pole, I will add such a proof below, just for the sake of completeness and curiosity of someone who would want to read it in the future. The proof is not extremely rigorous, since we're allowing the function to behave properly, evading complications as it was intended in the exercise. Also the proof is missing the picture of the contour, but one should have no problem drawing it with the directions given.

Regards!

We assume as below, that we have convergent Fourier series for ##f##, so that it can be integrated term by term, and we assume that the integrals in the exercise are convergent, that is, that on the left hand side we have Cauchy principal value of the integral.
Then:
$$f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos(2nx) + \sum_{n=1}^\infty b_n \sin(2nx)$$
From the convergence of the integral, we conclude that the sine series will give zero integrating term by term, since it is an odd function on a symmetric interval.
For the cosine part, the term we're integrating looks like:
$$\int_{-\infty}^{\infty} \cos(2nx) \frac{\tan(x)}{x}dx $$
Now we will search for principal value by using contour integration. Our contour will be anti-clockwise directed semicircle in the upper-half plane with infinite diameter along the real axis of the complex plane(a very standard contour, I don't have any good way to draw it here, so I assume the description of it is sufficient).
The function we will be integrating along this contour will be:
$$f(z) = e^{i(2nx)}\frac{\sin(z)}{z\cos(z)}$$
This function has simple poles at ##z = 0## and ##z = k\pi + \tfrac{\pi}{2}##, all along the real axis, so we will evade all of these poles by constructing small semi-circles in the upper-half plane around each pole, clockwise. This way, we won't have any poles inside the contour.
So, denoting the small semi-circles by ##C_0## and ##C_k##, the real axis part of the contour by ##\gamma##, the big semi-circle by ##C_R##, and the full contour by ##C##, we have:
$$\oint_C f(z)dz = \oint_{C_0} f(z)dz + \sum_{k=-m+1}^{m} \oint_{C_k}f(z)dz + \oint_{\gamma} f(z)dz + \oint_{C^+} f(z)dz$$
By Cauchy's residue theorem the total integral is zero since we have no poles inside the contour.
For the semicircles we will use Jordan's lemma. For now, we will assume the contour to be of finite width ##2R## with ##2m## poles in between(##k = -(m-1), \dots, m##), and in the end we will extend this contour to infinity.
For small semicircles, we have:
$$\lim_{r_k \rightarrow 0}\oint_{C_k} f(z)dz = -i\pi Res(f,k\pi+\tfrac{\pi}{2})$$
$$\lim_{r \rightarrow 0}\oint_{C_0} f(z)dz = -i\pi Res(f,0)$$
where we denoted the radii of the semicircles around the poles with ##r_k## and ##r##.

We calculate the residues:
$$Res(f,0) = \lim_{z \rightarrow 0} zf(z) = 0$$
$$Res(f,k\pi+\tfrac{\pi}{2}) = \lim_{z \rightarrow k\pi + \tfrac{\pi}{2}} \left(z - k\pi -\frac{\pi}{2}\right)f(z) = e^{i(2n(k\pi+\tfrac{\pi}{2}))}\sin\left(k\pi + \frac{\pi}{2}\right)\frac{2}{(2k+1)\pi} \lim_{z \rightarrow k\pi + \tfrac{\pi}{2}} \frac{z - k\pi - \frac{\pi}{2}}{\cos(z)} = (-1)^{(n+k)}\frac{2}{(2k+1)\pi} (-1)^{(k+1)} = \frac{2(-1)^{(n+1)}}{(2k+1)\pi}$$
Integral along the real axis(which we denoted by ##\gamma##) is:
$$\oint_{\gamma} f(z)dz = \int_{-R}^{R}e^{i(2nx)}\frac{\sin(x)}{x\cos(x)}dx$$
Combining the results above, we have:
$$ 0 = \oint_{C_R}f(z)dz + \int_{-R}^{R}e^{i(2nx)}\frac{\sin(x)}{x\cos(x)}dx +(-1)^n\sum_{k=-m+1}^{m} \frac{2i}{2k+1}$$
The integral we're looking for is the real part of the principal value we see in the expression above with boundaries going to infinity. When we take boundaries to infinity, ##C_R## integral will drop to zero, because by closing the contour anticlockwise in the upper-half of the plane, we obtained exponential damping of the integrand, so this limit will follow from Jordan's lemma. The sum that we have in the last term will diverge, however this sum is purely imaginary, the real part will still be zero. So we conclude that the real part of our integral is zero:
$$p.v. \int_{-\infty}^{\infty} \cos(2nx)\frac{\sin(x)}{x\cos(x)}dx = 0$$
This means that our cosine series will be integrated to zero term by term. We're left only with the constant term. So we have:
$$\int_{-\infty}^{\infty} f(x)\frac{\tan(x)}{x}dx = \frac{a_0}{2}\int_{\infty}^\infty \frac{\tan(x)}{x}dx$$
Where we're looking for the Cauchy principal value of the integral, as before. We will perform the same type of calculation as with the cosine series, this time working with the function:
$$f(z) = \frac{e^{iz}}{z\cos(z)}$$
The contour and method are completely identical, so we will proceed to calculate the residues:
$$Res(f,0) = 1$$
$$Res(f,k\pi + \frac{\pi}{2}) = \frac{-2i}{(2k+1)\pi}$$
From Cauchy integral theorem, we find:
$$0 = \oint_{C_R}f(z)dz -i\pi -\sum_{k=-m+1}^m \frac{2}{(2k+1)} + \int_{R}^{R} \frac{e^{ix}}{x\cos(x)}dx$$
Letting ##R## go to infinity, we find that the integral along the big semicircle ##C_R## goes to zero, analogous to the cosine-series case, and that the sum is divergent. However, the integral we're computing is imaginary part of the principal value in the formula above, and the sum is purely real. So we conclude:
$$p.v. \int_{-\infty}^{\infty} \frac{\tan(x)}{x}dx = \pi$$
Finally, we obtain:
$$p.v. \int_{-\infty}^{\infty} f(x)\frac{\tan(x)}{x}dx = \frac{\pi a_0}{2} = \int_0^\pi f(x)dx$$
So the correct identity is with Cauchy principal value on the left.
Wow! And I only used the power series of ##\operatorname{cot}(x-k\pi)##.
 
  • #61
137
62
The solution to this inequality draws inspiration from this integral:
$$\int \frac{dx}{1+k^2x^2} = \frac{1}{k}\arctan(kx) + C$$
We assume that the sequences on the right-hand side are well defined. Then we won't have any convergence problems.
We proceed using Cauchy-Schwarz inequality:
$$\left(\sum_{n \in \mathbb{N}}a_n\right)^2 \leq \sum_{n \in \mathbb{N}} a^2_n(1+k^2n^2)\sum_{n \in \mathbb{N}} \frac{1}{1+k^2n^2} $$.
Here ##k \in \mathbb{R}## is just a parameter, setting it up for the integral above.
The second sum can be bounded by the integral:
$$\sum_{n \in \mathbb{N}} \frac{1}{1+k^2n^2} < \int_0^\infty \frac{dx}{1+k^2x^2} = \frac{\pi}{2k}$$.
Combining the inequalities, we have:
$$\left(\sum_{n \in \mathbb{N}}a_n\right)^2 < \sum_{n \in \mathbb{N}} a^2_n(1+k^2n^2)\frac{\pi}{2k} = \frac{\pi}{2} \left(\frac{1}{k}\sum_{n \in \mathbb{N}} a^2_n + k \sum_{n \in \mathbb{N}} n^2a^2_n\right)$$
The term on the right is of the form ##\tfrac{x}{k} + yk##, and we need it to be of the form ##2\sqrt{xy}## in order to prove the inequality. So we will assume that it's of this form, and search for ##k## to confirm it, if possible. We find:
$$\frac{x}{k} + yk = 2\sqrt{xy} \Rightarrow \frac{x^2}{k^2} - 2xy + y^2k^2 = 0 \Rightarrow \left(\frac{x}{k} - yk\right) = 0 \Rightarrow k = \sqrt{\frac{x}{y}}$$.
So with choice:
$$k = \sqrt{\frac{\sum_{n \in \mathbb{N}} a^2_n}{\sum_{n \in \mathbb{N}} n^2a^2_n}}$$
we obtain the inequality that we're looking for by squaring both sides.
 
  • #62
12,533
8,948
The solution to this inequality draws inspiration from this integral:
$$\int \frac{dx}{1+k^2x^2} = \frac{1}{k}\arctan(kx) + C$$
We assume that the sequences on the right-hand side are well defined. Then we won't have any convergence problems.
We proceed using Cauchy-Schwarz inequality:
$$\left(\sum_{n \in \mathbb{N}}a_n\right)^2 \leq \sum_{n \in \mathbb{N}} a^2_n(1+k^2n^2)\sum_{n \in \mathbb{N}} \frac{1}{1+k^2n^2} $$.
Here ##k \in \mathbb{R}## is just a parameter, setting it up for the integral above.
The second sum can be bounded by the integral:
$$\sum_{n \in \mathbb{N}} \frac{1}{1+k^2n^2} < \int_0^\infty \frac{dx}{1+k^2x^2} = \frac{\pi}{2k}$$.
Combining the inequalities, we have:
$$\left(\sum_{n \in \mathbb{N}}a_n\right)^2 < \sum_{n \in \mathbb{N}} a^2_n(1+k^2n^2)\frac{\pi}{2k} = \frac{\pi}{2} \left(\frac{1}{k}\sum_{n \in \mathbb{N}} a^2_n + k \sum_{n \in \mathbb{N}} n^2a^2_n\right)$$
The term on the right is of the form ##\tfrac{x}{k} + yk##, and we need it to be of the form ##2\sqrt{xy}## in order to prove the inequality. So we will assume that it's of this form, and search for ##k## to confirm it, if possible. We find:
$$\frac{x}{k} + yk = 2\sqrt{xy} \Rightarrow \frac{x^2}{k^2} - 2xy + y^2k^2 = 0 \Rightarrow \left(\frac{x}{k} - yk\right) = 0 \Rightarrow k = \sqrt{\frac{x}{y}}$$.
So with choice:
$$k = \sqrt{\frac{\sum_{n \in \mathbb{N}} a^2_n}{\sum_{n \in \mathbb{N}} n^2a^2_n}}$$
we obtain the inequality that we're looking for by squaring both sides.
The inequality is called Carlson's inequality. Hardy gave two proofs, one with the integration trick and Schwarz's inequality as above, and another which uses Parseval's inequality (see November 2019 challenge #1).

So if anyone wants to search for the second proof, go ahead!
 

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