SUMMARY
The inverse function of the hyperbolic tangent, defined as ##f:= \tanh(x) = \frac{e^x-e^{-x}}{e^x+e^{-x}}##, is proven to be ##f^{-1}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)## for all x in the interval (-1, 1). Additionally, the series representation of the inverse function is given by ##f^{-1}(x) = \sum\limits_{k=0}^{\infty} \frac{x^{2k+1}}{2k+1}##. The discussion highlights the relevance of the Taylor expansion for logarithmic functions, specifically ##\ln(1+x)## and ##\ln(1-x)##, in deriving these results.
PREREQUISITES
- Understanding of hyperbolic functions, specifically the hyperbolic tangent function.
- Familiarity with logarithmic functions and their properties.
- Knowledge of Taylor series expansions and their applications.
- Basic calculus concepts, including limits and convergence of series.
NEXT STEPS
- Study the Taylor series expansion of ##\ln(1+x)## and ##\ln(1-x)##.
- Explore the properties and applications of hyperbolic functions in calculus.
- Investigate the convergence criteria for power series within specific intervals.
- Learn about the relationship between inverse functions and their derivatives.
USEFUL FOR
Students studying calculus, mathematicians interested in hyperbolic functions, and anyone seeking to understand the properties of inverse functions within the context of real analysis.