Prove: Integral Of Arctan²(x)arctanh(x²)/x = G²

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Hey guys! Let's dive into a fascinating integral problem that pops up in the realms of real analysis and calculus. We're going to explore how to prove this intriguing identity:

$$\operatorname{\mathfrak{R}} \left\{\int _0^{\infty }\frac{\arctan ^2\left(x\right)\operatorname{arctanh} \left(x^2\right)}{x}\,dx\right\}=G^2,$$

where G represents Catalan's constant. This journey will take us through the clever application of various integration techniques and a bit of complex analysis. So, buckle up, and let’s get started!

Understanding the Integral and the Goal

Before we jump into the nitty-gritty details, let's break down what we're dealing with. The integral in question involves the product of arctan²(x) and arctanh(x²), all divided by x. Our mission, should we choose to accept it (and we do!), is to show that the real part of this definite integral from 0 to infinity equals G², where G is Catalan's constant.

Catalan's constant, denoted by G, is a special constant that appears in various areas of mathematics, particularly in combinatorics and number theory. It's defined by the following infinite series:

$$G = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^2} = 1 - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots$$

Its approximate numerical value is around 0.915965594. Now that we know what we're aiming for, let's strategize how to get there.

Strategic Approaches to Tackle the Integral

To solve this integral, we'll employ a combination of techniques, including:

1. Integration by parts: This is a classic technique that can help us simplify the integral by transferring the derivative from one part of the integrand to another.
2. Series representation: Expressing arctanh(x²) as an infinite series can help us break down the complex function into simpler terms.
3. Substitution: A well-chosen substitution can often transform an integral into a more manageable form.
4. Complex analysis (optional): While not strictly necessary, complex analysis can provide elegant solutions to certain integrals.

Let's start by exploring the series representation of arctanh(x²).

Leveraging the Series Representation of arctanh(x²)

The function arctanh(x) has a well-known series representation given by:

$$\operatorname{arctanh}(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{2n+1}, \quad |x| < 1$$

Replacing x with x², we get:

$$\operatorname{arctanh}(x^2) = \sum_{n=0}^{\infty} \frac{x^{4n+2}}{2n+1}, \quad |x| < 1$$

This series representation is valid for |x| < 1. However, we're integrating from 0 to infinity. This means we'll need to be careful about convergence and potentially use analytic continuation if we want to extend our results to the entire range of integration.

For now, let’s focus on the series representation and plug it into our integral. This gives us:

$$\int _0^{\infty }\frac{\arctan ^2(x)}{x} \sum_{n=0}^{\infty} \frac{x^{4n+2}}{2n+1} dx$$

Now, we want to interchange the summation and integration. This step requires careful justification, as it's not always valid to do so. However, for the sake of exploration, let's assume we can interchange them and see where it leads us. This gives us:

$$\sum_{n=0}^{\infty} \frac{1}{2n+1} \int _0^{\infty } x^{4n+1} \arctan ^2(x) dx$$

Now we have a family of integrals to tackle: $\int _0^{\infty } x^{4n+1} \arctan ^2(x) dx$. This looks a bit more manageable, but we still need a strategy to evaluate it.

Tackling the Integral $\int _0^{\infty } x^{4n+1} \arctan ^2(x) dx$ with Integration by Parts

To evaluate the integral $\int _0^{\infty } x^{4n+1} \arctan ^2(x) dx$, we can use integration by parts. Recall the integration by parts formula:

$$\int u dv = uv - \int v du$$

Let's choose:

• u = arctan²(x), so du = (2 arctan(x))/(1 + x²) dx
• dv = x^(4n+1) dx, so v = x^(4n+2) / (4n+2)

Applying integration by parts, we get:

$$\int _0^{\infty } x^{4n+1} \arctan ^2(x) dx = \left[ \frac{x^{4n+2}}{4n+2} \arctan ^2(x) \right]_0^{\infty } - \int _0^{\infty } \frac{x^{4n+2}}{4n+2} \frac{2 \arctan(x)}{1 + x^2} dx$$

The first term, $\left[ \frac{x^{4n+2}}{4n+2} \arctan ^2(x) \right]_0^{\infty }$, requires careful evaluation of the limits. As x approaches infinity, arctan²(x) approaches (π/2)², but x^(4n+2) also goes to infinity. This suggests we might need to use L'Hôpital's rule or a similar technique to evaluate this limit properly. However, for now, let's assume that this term evaluates to zero (we'll need to verify this rigorously later). This leaves us with:

$$- \frac{2}{4n+2} \int _0^{\infty } \frac{x^{4n+2} \arctan(x)}{1 + x^2} dx = - \frac{1}{2n+1} \int _0^{\infty } \frac{x^{4n+2} \arctan(x)}{1 + x^2} dx$$

We now have a new integral to tackle: $\int _0^{\infty } \frac{x^{4n+2} \arctan(x)}{1 + x^2} dx$. This integral looks challenging, but let's try another round of integration by parts.

A Second Round of Integration by Parts

Let's apply integration by parts again to the integral $\int _0^{\infty } \frac{x^{4n+2} \arctan(x)}{1 + x^2} dx$. This time, let's choose:

• u = arctan(x), so du = 1/(1 + x²)* dx
• dv = x^(4n+2) / (1 + x²) dx

However, finding v in this case is not straightforward. We might need to use a different approach or a clever substitution to simplify this integral. Another avenue we can explore is to use a trigonometric substitution or to relate this integral to known integrals involving arctangent.

The Clever Substitution: x = tan(θ)

A substitution that often works well with arctangent integrals is x = tan(θ). This substitution gives us dx = sec²(θ) dθ, and arctan(x) = θ. The limits of integration change from 0 to π/2. Let's apply this substitution to the integral $\int _0^{\infty } \frac{x^{4n+2} \arctan(x)}{1 + x^2} dx$:

$$\int _0^{\infty } \frac{x^{4n+2} \arctan(x)}{1 + x^2} dx = \int _0^{\frac{\pi}{2}} \frac{\tan^{4n+2}(\theta) \cdot \theta}{1 + \tan^2(\theta)} \sec^2(\theta) d\theta$$

Since 1 + tan²(θ) = sec²(θ), the sec²(θ) terms cancel out, leaving us with:

$$\int _0^{\frac{\pi}{2}} \tan^{4n+2}(\theta) \cdot \theta \:d\theta$$

This integral looks more manageable. However, evaluating $\int _0^{\frac{\pi}{2}} \tan^{4n+2}(\theta) \theta \:d\theta$ can still be tricky. We might need to use further integration techniques or try to relate it to known integrals.

Connecting to Known Integrals and Catalan's Constant

At this point, we've made some progress, but we're not quite at the finish line. We need to find a way to connect the integral $\int _0^{\frac{\pi}{2}} \tan^{4n+2}(\theta) \theta \:d\theta$ back to Catalan's constant G. This often involves looking for patterns and using known integral identities.

One possible strategy is to try to express the integral in terms of polylogarithm functions or other special functions that are known to be related to Catalan's constant. Another approach could be to explore complex analysis techniques, such as contour integration, to evaluate the integral.

A Glimpse into a Potential Solution Path

While a complete, step-by-step solution can be quite involved and lengthy, let’s sketch out a potential path that leverages some advanced techniques:

1. Express the integral in terms of a series: We’ve already started this by using the series representation of arctanh(x²).
2. Evaluate the resulting integrals: This is the trickiest part and might involve a combination of integration by parts, substitutions, and complex analysis.
3. Relate the result to Catalan's constant: This often involves recognizing specific series or integral representations of G.

For instance, integrals of the form $\int _0^{\frac{\pi}{2}} \theta \tan^m(\theta) d\theta$ can sometimes be evaluated using complex analysis or by relating them to the derivatives of certain special functions.

The Power of Collaboration and Further Exploration

This integral is a great example of how challenging definite integrals can be. It often requires a blend of different techniques and a bit of ingenuity to arrive at the solution. The journey we've taken so far highlights the key strategies and the potential roadblocks we might encounter.

Solving this integral completely would likely involve diving deeper into complex analysis, exploring specific integral identities, and possibly using computer algebra systems to help with the calculations. It's a problem that could benefit from collaboration and further exploration.

Conclusion: The Beauty of Integral Problems

While we haven't presented a complete, step-by-step solution here, we've explored the main ideas and techniques that can be used to tackle this fascinating integral. The identity

$$\operatorname{\mathfrak{R}} \left\{\int _0^{\infty }\frac{\arctan ^2\left(x\right)\operatorname{arctanh} \left(x^2\right)}{x}\,dx\right\}=G^2$$

is a testament to the beauty and complexity of integral calculus. It showcases how different areas of mathematics, such as real analysis, complex analysis, and special functions, can come together to solve a single problem. Keep exploring, keep questioning, and keep integrating! You got this!