Proof Of The Inverse Hyperbolic Tangent Identity

In the realm of mathematics, elegant equations often conceal profound relationships. One such equation is the identity for the inverse hyperbolic tangent function, which connects it to the natural logarithm. This article delves into a rigorous proof of this identity, providing a step-by-step derivation that illuminates the underlying mathematical principles. Our main focus will be on proving the following equation using a method similar to the provided example:

$\tanh ^{-1}(x)=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right) \quad-1 < x < 1$

This equation reveals a surprising connection between the inverse hyperbolic tangent function, denoted as $\tanh^{-1}(x)$, and the natural logarithm function, represented by $\ln$. The condition $-1 < x < 1$ is crucial as it defines the domain for which the inverse hyperbolic tangent function is well-defined. In this exploration, we will embark on a journey to unravel the intricacies of this identity, providing a clear and concise proof that showcases the beauty and interconnectedness of mathematical concepts.

Laying the Foundation: Defining the Inverse Hyperbolic Tangent

To embark on our proof, it's imperative to first establish a firm understanding of the inverse hyperbolic tangent function. The inverse hyperbolic tangent, denoted as $\tanh^{-1}(x)$, is the inverse function of the hyperbolic tangent function, $\tanh(x)$. In simpler terms, if $y = \tanh^{-1}(x)$, then $x = \tanh(y)$. This fundamental relationship forms the cornerstone of our proof. We can express the hyperbolic tangent function in terms of exponential functions, which is crucial for bridging the gap between hyperbolic functions and logarithms.

Recall that the hyperbolic tangent function is defined as the ratio of the hyperbolic sine function to the hyperbolic cosine function:

$\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$

Furthermore, the hyperbolic sine and cosine functions are defined in terms of exponential functions as follows:

$\sinh(x) = \frac{e^x - e^{-x}}{2}$

$\cosh(x) = \frac{e^x + e^{-x}}{2}$

Substituting these definitions into the expression for $\tanh(x)$, we obtain:

$\tanh(x) = \frac{\frac{e^x - e^{-x}}{2}}{\frac{e^x + e^{-x}}{2}} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$

This exponential form of the hyperbolic tangent function is the key to unlocking the identity we aim to prove. By manipulating this expression and leveraging the properties of logarithms, we can establish the equivalence between $\tanh^{-1}(x)$ and the logarithmic expression in the identity. The interplay between exponential and logarithmic functions is a recurring theme in mathematics, and this proof beautifully illustrates this connection.

The Proof Unveiled: A Step-by-Step Derivation

Now, let's dive into the heart of the proof. To demonstrate the identity $\tanh ^{-1}(x)=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$, we will follow a logical sequence of steps, starting with the definition of the inverse hyperbolic tangent and culminating in the desired logarithmic expression.

1. Begin with the Definition: Let $y = \tanh^{-1}(x)$. This is our starting point, establishing a relationship between $y$ and $x$ through the inverse hyperbolic tangent function.

2. Express in terms of Hyperbolic Tangent: From the definition of the inverse function, we can rewrite the equation as $x = \tanh(y)$. This step transforms the inverse relationship into a direct relationship, making it easier to work with.

3. Substitute the Exponential Form: Replace $\tanh(y)$ with its exponential definition: $x = \frac{e^y - e^{-y}}{e^y + e^{-y}}$. This is a crucial step that introduces exponential functions into the equation, paving the way for logarithmic manipulation. The exponential form allows us to express the hyperbolic tangent in terms of more fundamental functions.

4. Algebraic Manipulation: Multiply both the numerator and denominator of the right-hand side by $e^y$: $x = \frac{e^{2y} - 1}{e^{2y} + 1}$. This manipulation simplifies the expression and brings us closer to isolating the exponential term. The strategic multiplication by $e^y$ eliminates the negative exponent and allows us to work with a cleaner expression.

5. Isolate the Exponential Term: Rearrange the equation to solve for $e^{2y}$. This involves a series of algebraic steps: multiply both sides by $(e^{2y} + 1)$, then isolate the terms with $e^{2y}$ on one side and the constant terms on the other. This process leads to the equation: $e^{2y} = \frac{1 + x}{1 - x}$. Isolating the exponential term is a key step in preparing to apply the natural logarithm.

6. Apply the Natural Logarithm: Take the natural logarithm of both sides of the equation: $\ln(e^{2y}) = \ln\left(\frac{1 + x}{1 - x}\right)$. The natural logarithm is the inverse function of the exponential function, so applying it to $e^{2y}$ will help us isolate $y$. This step leverages the fundamental relationship between exponential and logarithmic functions.

7. Simplify: Simplify the left-hand side using the property of logarithms that $\ln(e^a) = a$: $2y = \ln\left(\frac{1 + x}{1 - x}\right)$. This simplification brings us closer to our goal of expressing $y$ in terms of a logarithmic expression.

8. Solve for y: Divide both sides by 2 to isolate $y$: $y = \frac{1}{2} \ln\left(\frac{1 + x}{1 - x}\right)$. This step completes the derivation, expressing $y$ in the desired form.

9. Substitute Back: Recall that we initially defined $y = \tanh^{-1}(x)$. Substitute this back into the equation to obtain the final result: $\tanh^{-1}(x) = \frac{1}{2} \ln\left(\frac{1 + x}{1 - x}\right)$.

Thus, we have successfully proven the identity for the inverse hyperbolic tangent function. Each step in this derivation was carefully chosen to transform the initial definition into the desired logarithmic expression. The combination of algebraic manipulation, exponential forms, and logarithmic properties is a testament to the power of mathematical techniques.

Domain Considerations: The Importance of -1 < x < 1

As mentioned earlier, the identity $\tanh ^{-1}(x)=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$ holds true only for $-1 < x < 1$. This restriction arises from the domain of the inverse hyperbolic tangent function and the argument of the natural logarithm.

The inverse hyperbolic tangent function, $\tanh^{-1}(x)$, is defined only for values of $x$ within the interval $(-1, 1)$. This is because the range of the hyperbolic tangent function, $\tanh(x)$, is $(-1, 1)$. For a function to have an inverse, its range must become the domain of the inverse function. Therefore, $\tanh^{-1}(x)$ can only accept inputs between -1 and 1.

Furthermore, the natural logarithm function, $\ln(x)$, is defined only for positive arguments. In the identity, the argument of the natural logarithm is $\frac{1+x}{1-x}$. For this expression to be positive, both the numerator and denominator must have the same sign. Let's analyze the conditions:

• If $1 + x > 0$ and $1 - x > 0$, then $x > -1$ and $x < 1$, which gives us the interval $-1 < x < 1$.
• If $1 + x < 0$ and $1 - x < 0$, then $x < -1$ and $x > 1$, which is an impossible condition.

Therefore, the condition $-1 < x < 1$ is necessary to ensure that the argument of the natural logarithm is positive and that the inverse hyperbolic tangent function is defined. This domain restriction is a crucial aspect of the identity and must be considered when applying it.

Applications and Significance of the Identity

The identity $\tanh ^{-1}(x)=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$ is not merely a theoretical curiosity; it has practical applications in various fields, including physics, engineering, and computer science. Understanding this identity allows us to convert between inverse hyperbolic tangent expressions and logarithmic expressions, which can simplify calculations and provide alternative perspectives on problems.

In physics, hyperbolic functions and their inverses often appear in the context of relativity and electromagnetism. The inverse hyperbolic tangent identity can be used to simplify expressions involving velocities and rapidities in relativistic calculations. It can also be applied in the analysis of transmission lines and wave propagation in electrical engineering.

In computer science, hyperbolic functions and their inverses find applications in machine learning and neural networks. The inverse hyperbolic tangent function is sometimes used as an activation function in neural networks, and the identity can be helpful in analyzing and optimizing the performance of these networks.

Beyond these specific applications, the identity serves as a beautiful example of the interconnectedness of mathematical concepts. It highlights the relationship between hyperbolic functions, exponential functions, and logarithmic functions, demonstrating how seemingly disparate areas of mathematics are intimately related. The ability to connect different mathematical concepts is a hallmark of mathematical understanding and a key to solving complex problems.

Conclusion: A Testament to Mathematical Elegance

In conclusion, we have successfully proven the identity $\tanh ^{-1}(x)=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$ for $-1 < x < 1$. This proof involved a careful sequence of steps, starting with the definition of the inverse hyperbolic tangent function and culminating in the desired logarithmic expression. We emphasized the importance of the domain restriction $-1 < x < 1$ and discussed the applications and significance of the identity in various fields.

This identity stands as a testament to the elegance and interconnectedness of mathematics. It showcases the power of algebraic manipulation, exponential forms, and logarithmic properties in establishing fundamental relationships between mathematical functions. By understanding and appreciating such identities, we deepen our understanding of the mathematical world and equip ourselves with the tools to tackle complex problems in various domains. The beauty of mathematics lies not only in its practical applications but also in its inherent elegance and logical structure.