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:

    “Let’s just call what happened in the eighties the reclamation of motherhood . . . by women I knew and loved, hard-driving women with major careers who were after not just babies per se or motherhood per se, but after a reconciliation with their memories of their own mothers. So having a baby wasn’t just having a baby. It became a major healing.”
    —Anne Taylor Fleming (20th century)

    “I thought I never wanted to be a father. A child seemed to be a series of limitations and responsibilities that offered no reward. But when I experienced the perfection of fatherhood, the rest of the world remade itself before my eyes.”
    —Kent Nerburn (20th century)

    “Its idea of “production value” is spending a million dollars dressing up a story that any good writer would throw away. Its vision of the rewarding movie is a vehicle for some glamour-puss with two expressions and eighteen changes of costume, or for some male idol of the muddled millions with a permanent hangover, six worn-out acting tricks, the build of a lifeguard, and the mentality of a chicken-strangler.”
    —Raymond Chandler (1888–1959)