A hard integral gives a simple closed form, π/(4a)^3

In summary, a hard integral is a difficult integral to solve using traditional algebraic methods and may require advanced techniques. A simple closed form refers to a mathematical expression that can be written using a finite number of basic operations and functions. Some hard integrals have known solutions in simple closed form, including the value π/(4a)^3, which serves as an example of how a seemingly complex problem can be simplified. However, not all hard integrals have solutions in simple closed form and may require numerical methods or have solutions that cannot be expressed using known functions.
  • #1
Tony1
17
0
Proposed:

How can we prove $(1)?$

$$\int_{0}^{\infty}\mathrm dx{\sin^2\left({a\over x}\right)\over (4a^2+x^2)^2}={\pi\over (4a)^3}\tag1$$

We can start to decompose $(1)$ to...

$$\int_{0}^{\infty}\mathrm dx{1\over (4a^2+x^2)^2}-\int_{0}^{\infty}\mathrm dx{\cos^2\left({a\over x}\right)\over (4a^2+x^2)^2}\tag2$$

$$\int_{0}^{\infty}\mathrm dx{1\over (4a^2+x^2)^2}-{1\over 2}\int_{0}^{\infty}\mathrm dx{1\over (4a^2+x^2)^2}-{1\over 2}\int_{0}^{\infty}\mathrm dx{\cos\left({2a\over x}\right)\over (4a^2+x^2)^2}\tag3$$

$${1\over 2}\int_{0}^{\infty}\mathrm dx{1\over (4a^2+x^2)^2}-{1\over 2}\int_{0}^{\infty}\mathrm dx{\cos\left({2a\over x}\right)\over (4a^2+x^2)^2}\tag4$$

So far...
 
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  • #2
Hi, Tony.

Your work looks good so far. Next, you can use a $u$-substitution to show that
$$\int_{0}^{\infty}\frac{\cos\left(\frac{2a}{x}\right)}{(4a^{2}+x^{2})^{2}}dx=\frac{1}{(2a)^{3}}\int_{0}^{\infty}\frac{x^{2}\cos (x)}{(x^{2}+1)^{2}}dx.$$
The idea is to now notice that the two integrals in (4) --i.e.,
$$\int_{0}^{\infty}\frac{1}{(4a^{2}+x^{2})^{2}}dx\qquad\text{and}\qquad\int_{0}^{\infty}\frac{x^{2}\cos (x)}{(x^{2}+1)^{2}}dx$$
-- are even functions of $x$, so can be expressed as
$$\frac{1}{2}\int_{-\infty}^{\infty}\frac{1}{(4a^{2}+x^{2})^{2}}dx\qquad\text{and}\qquad\frac{1}{2}\int_{-\infty}^{\infty}\frac{x^{2}\cos (x)}{(x^{2}+1)^{2}}dx.$$
Now use the method of contour integration on these integrals to obtain the desired result.
 

FAQ: A hard integral gives a simple closed form, π/(4a)^3

1. What is a hard integral?

A hard integral is an integral that is difficult to solve using traditional algebraic methods. It may require advanced techniques such as substitution, integration by parts, or trigonometric identities to find a solution.

2. What does it mean for an integral to give a simple closed form?

A simple closed form refers to a mathematical expression that can be written using a finite number of basic operations and functions. In the context of integrals, a simple closed form means that the solution can be expressed using a finite combination of known functions, such as polynomials, trigonometric functions, or logarithms.

3. How can a hard integral give a simple closed form?

There are certain integrals that, despite being difficult to solve, have well-known solutions that can be expressed in simple, closed forms. These solutions have been discovered through years of mathematical research and provide a useful shortcut for solving difficult problems.

4. What is the significance of the value π/(4a)^3 in this context?

The value π/(4a)^3 represents the solution to a specific hard integral that has been solved using advanced mathematical techniques. It serves as an example of how a seemingly complex problem can be simplified and solved using a simple closed form.

5. Can all hard integrals be solved using simple closed forms?

No, not all hard integrals have known solutions in simple closed form. Some integrals can only be approximated using numerical methods, while others may have solutions that cannot be expressed in terms of known functions.

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