Formulae for the exponential, the hyperbolic and the trigonometric functions in terms of the logarithmic function

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Ioakimidis, Nikolaos
Anastasselou, Eleni
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A common definition of the exponential function is as the inverse function of the logarithmic function, which is defined as the definite integral of the rational function 1/t over the interval [1,x] with x > 0. The hyperbolic functions (hyperbolic sine, cosine, tangent, etc.) are next defined in terms of the exponential function. Here we derive an explicit real formula for the hyperbolic tangent function in terms of the logarithmic function, which is sufficient for the direct derivation of analogous formulae for the exponential function and the other hyperbolic functions. A similar formula for the trigonometric tangent function, which can be directly used for the derivation of analogous formulae for the other trigonometric functions, is also derived. The present results are based on a simple method for the derivation of closed-form formulae for the zeros of sectionally analytic functions.
Exponential function, Hyperbolic functions, Trigonometric functions, Logarithmic function, Elementary transcendental functions, Inverse functions, Closed-form formulae, Sectionally analytic functions, Sectionally meromorphic functions, Zeros and poles