Hyperbolic Function - Taylor Series Expressions

Taylor Series Expressions

It is possible to express the above functions as Taylor series:

The function sinh x has a Taylor series expression with only odd exponents for x. Thus it is an odd function, that is, −sinh x = sinh(−x), and sinh 0 = 0.

The function cosh x has a Taylor series expression with only even exponents for x. Thus it is an even function, that is, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the infinite series expression of the exponential function.

\begin{align} \tanh x &= x - \frac {x^3} {3} + \frac {2x^5} {15} - \frac {17x^7} {315} + \cdots = \sum_{n=1}^\infty \frac{2^{2n}(2^{2n}-1)B_{2n} x^{2n-1}}{(2n)!}, \left |x \right | < \frac {\pi} {2} \\ \coth x &= x^{-1} + \frac {x} {3} - \frac {x^3} {45} + \frac {2x^5} {945} + \cdots = x^{-1} + \sum_{n=1}^\infty \frac{2^{2n} B_{2n} x^{2n-1}} {(2n)!}, 0 < \left |x \right | < \pi \\ \operatorname {sech}\, x &= 1 - \frac {x^2} {2} + \frac {5x^4} {24} - \frac {61x^6} {720} + \cdots = \sum_{n=0}^\infty \frac{E_{2 n} x^{2n}}{(2n)!}, \left |x \right | < \frac {\pi} {2} \\ \operatorname {csch}\, x &= x^{-1} - \frac {x} {6} +\frac {7x^3} {360} -\frac {31x^5} {15120} + \cdots = x^{-1} + \sum_{n=1}^\infty \frac{ 2 (1-2^{2n-1}) B_{2n} x^{2n-1}}{(2n)!}, 0 < \left |x \right | < \pi \end{align}

where

is the nth Bernoulli number
is the nth Euler number

Read more about this topic:  Hyperbolic Function

Famous quotes containing the words taylor, series and/or expressions:

    “A religion, that is, a true religion, must consist of ideas and facts both; not of ideas alone without facts, for then it would be mere Philosophy;Mnor of facts alone without ideas, of which those facts are symbols, or out of which they arise, or upon which they are grounded: for then it would be mere History.”
    —Samuel Taylor Coleridge (1772–1834)

    “I look on trade and every mechanical craft as education also. But let me discriminate what is precious herein. There is in each of these works an act of invention, an intellectual step, or short series of steps taken; that act or step is the spiritual act; all the rest is mere repetition of the same a thousand times.”
    —Ralph Waldo Emerson (1803–1882)

    “Our books are false by being fragmentary: their sentences are bon mots, and not parts of natural discourse; childish expressions of surprise or pleasure in nature; or, worse, owing a brief notoriety to their petulance, or aversion from the order of nature,—being some curiosity or oddity, designedly not in harmony with nature, and purposely framed to excite surprise, as jugglers do by concealing their means.”
    —Ralph Waldo Emerson (1803–1882)