Question

Prove that the inverse hyperbolic functions are the following logarithms: a. \(\cosh ^{-1} x=\ln \left(x+\sqrt{x^{2}-1}\right) .\) b. \(\tanh ^{-1} x=\frac{1}{2} \ln \frac{1+x}{1-x}\).

Short answer

The expressions are true: \( \cosh ^{-1} x=\ln \left(x+\sqrt{x^{2}-1}\right) \) and \( \tanh ^{-1} x=\frac{1}{2} \ln \frac{1+x}{1-x} \).

Step-by-step solution

Use the definition of inverse function

Firstly, we should know that if \( y = \cosh^{-1}x \), then \( x = \cosh y \). And if \( y = \tanh^{-1}x \), then \( x = \tanh y \). This will serve as our working equations.

Step 2(a): Rewrite the hyperbolic cosine in terms of exponentials

The equation \( x = \cosh y \) can be expanded into exponential form as \( x = \frac{e^y + e^{-y}}{2} \).

Step 3(a): Solve the equation for y

Now we create a single-value equation by getting rid of \( e^{-y} \). Multiply through by \( e^y \) and rearrange to get quadratical equation \( e^{2y} - 2xe^{y} + 1 = 0 \). Solve that using the Quadratic formula brings us to \( e^y = x \pm \sqrt{x^2 - 1} \). Since \( e^y > 0 \) for all real \( y \), we choose the positive root, producing \( e^y = x + \sqrt{x^2 - 1} \). Therefore we reach \( y = \cosh^{-1}x = \ln(x + \sqrt{x^2 - 1}) \).

Step 2(b): Rewrite the hyperbolic tangent in terms of exponentials

By definition, \( x = \tanh y \) equals \( x = \frac{e^{2y} - 1}{e^{2y} + 1} \).

Step 3(b): Solve the equation for y

For the equation \( x = \frac{e^{2y} - 1}{e^{2y} + 1} \), solve for \( e^{2y} \) and let \( u = e^{2y} \). Then we get \( u = \frac{1 + x}{1 - x} \). Thus, via natural logarithm, we reach \( y = \tanh^{-1}x = \frac{1}{2} \ln \frac{1 + x}{1 - x} \).

Key concepts

Hyperbolic Functions

Hyperbolic functions are the hyperbolic analogs of the ordinary trigonometric, or circular, functions. Just as trigonometric functions can be defined as ratios of the sides of a right triangle or as points on the unit circle, hyperbolic functions relate to hyperbolas. The most important hyperbolic functions are the hyperbolic sine \( \sinh \) and the hyperbolic cosine \( \cosh \), from which others can be derived, including the hyperbolic tangent \( \tanh \).

These functions have similar properties to their trigonometric counterparts but differ in certain aspects due to their underlying relationship with the exponential function \( e^x \). For example, the hyperbolic cosine \( \cosh(x) \) is defined as \( \frac{e^x + e^{-x}}{2} \), and the hyperbolic sine \( \sinh(x) \) is defined as \( \frac{e^x - e^{-x}}{2} \). These definitions demonstrate a fundamental trait of hyperbolic functions: They can be written in terms of exponential functions, which makes them invaluable when solving certain types of equations.

Natural Logarithms

The natural logarithm, denoted by \( \ln \), is a logarithm with the special base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm maps multiplication into addition—more precisely, it converts the relationship between the base \( e \) and its exponent into a linear relationship.

For instance, if we have \( e^x = y \) for some real number \( y \) greater than zero, then \( x = \ln(y) \). This conversion is crucial in many mathematical calculations, including the solving of exponential equations and the integration of functions. When dealing with inverse hyperbolic functions, the natural logarithm is essential in 'unwrapping' the complexity of the relationships and reducing them to expressions that are more familiar and workable.

Exponential Functions

Exponential functions are mathematical expressions that describe growth or decay processes, with the general form \( y = a^x \) where \( a \) is a positive constant. The most prevalent exponential function is of the base \( e \)—commonly denoted as \( y = e^x \).

The exponential function with base \( e \) has a unique property; its rate of growth is proportional to its current value. This makes \( e^x \) an extremely important function in calculus, particularly for describing continuously changing phenomena.

Mathematically, the power of \( e^x \) lies in its derivative and integral, as the derivative of \( e^x \) is \( e^x \) itself, and likewise, its integral also remains \( e^x \) (up to a constant of integration). These properties simplify many calculus problems, and understanding the behavior of the exponential function is key to solving a range of equations, including hyperbolic functions.