Show that tanh^-1 x=(1)/(2) ln((1+x)/(1-x)), -1<x<1 |...

Name: Problem 61, Chapter 3: Calculus, Early Transcendentals
Uploaded: 2021-08-02T09:46:01-08:00
Duration: 2 min 48 s
Channel: Umar Sohail Qureshi
Description: Show that tanh^-1 x=(1)/(2) ln((1+x)/(1-x)), -1<x<1

Key Concepts

Domain and Range Considerations

Considering the domain and range of functions is crucial, especially for inverse functions, to ensure that the expressions are mathematically valid. For the inverse hyperbolic tangent, it is important to note that the function is defined only for inputs between -1 and 1, which ensures that the logarithmic representation is valid and that the inverse function remains one-to-one.

Logarithmic Representations

Logarithmic representations of inverse hyperbolic functions provide a powerful method to convert complex hyperbolic expressions into forms that are more manageable algebraically. Specifically, expressing the inverse hyperbolic tangent in terms of a natural logarithm simplifies calculations and highlights the deep connection between hyperbolic functions and exponential/logarithmic behavior.

Hyperbolic Tangent Function

The hyperbolic tangent function, defined as the ratio of the hyperbolic sine and hyperbolic cosine functions, exhibits properties analogous to the regular tangent function but in a hyperbolic context. Its behavior, range, and symmetry are essential when formulating its inverse and understanding the relationship between hyperbolic and logarithmic forms.

Inverse Hyperbolic Functions

Inverse hyperbolic functions are the inverses of the hyperbolic functions, allowing one to express angles in terms of ratios of hyperbolic sine and cosine. They are important in many branches of mathematics and physics for solving equations involving hyperbolic functions and provide a connection between algebraic expressions and geometric or analytic interpretations.

Transcript

Please give Ace some feedback