Hyperbolic secant function
This functionality does not run in MATLAB.
sech(x) represents the hyperbolic secant function 1/cosh(x).
This function is defined for complex arguments.
Floating point values are returned for floating-point arguments. Floating point intervals are returned for floating-point interval arguments. Unevaluated function calls are returned for most exact arguments.
Arguments that are integer multiples of lead to simplified results. If the argument involves a negative numerical factor of Type::Real, then symmetry relations are used to make this factor positive. Cf. Example 2.
sech(x) is rewritten as 1/cosh(x). Cf. Example 4.
The inverse function is implemented by arcsech. Cf. Example 5.
The float attributes are kernel functions, i.e., floating-point evaluation is fast.
When called with a floating-point argument, the functions are sensitive to the environment variable DIGITS which determines the numerical working precision.
We demonstrate some calls with exact and symbolic input data:
sinh(I*PI), cosh(1), tanh(5 + I), csch(PI), sech(1/11), coth(8)
sinh(x), cosh(x + I*PI), tanh(x^2 - 4)
Floating point values are computed for floating-point arguments:
sinh(123.4), cosh(5.6 + 7.8*I), coth(1.0/10^20)
For floating-point intervals, intervals enclosing the image are calculated:
cosh(-1 ... 1), tanh(-1 ... 1)
For functions with discontinuities, evaluation over an interval may result in a union of intervals:
coth(-1 ... 1)
Simplifications are implemented for arguments that are integer multiples of :
sinh(I*PI/2), cosh(40*I*PI), tanh(-10^100*I*PI), coth(-17/2*I*PI)
Negative real numerical factors in the argument are rewritten via symmetry relations:
sinh(-5), cosh(-3/2*x), tanh(-x*PI/12), coth(-12/17*x*y*PI)
The expand function implements the addition theorems:
expand(sinh(x + PI*I)), expand(cosh(x + y))
The combine function uses these theorems in the other direction, trying to rewrite products of hyperbolic functions:
Various relations exist between the hyperbolic functions:
Use rewrite to obtain a representation in terms of a specific target function:
rewrite(tanh(x)*exp(2*x), sinhcosh), rewrite(sinh(x), tanh)
rewrite(sinh(x)*coth(y), exp), rewrite(exp(x), coth)
sinh(arcsinh(x)), sinh(arccosh(x)), cosh(arctanh(x))
Note that arcsinh(sinh(x)) does not necessarily yield x, because arcsinh produces values with imaginary parts in the interval :
arcsinh(sinh(3)), arcsinh(sinh(1.6 + 100*I))
diff(sinh(x^2), x), float(sinh(3)*coth(5 + I))
limit(x*sinh(x)/tanh(x^2), x = 0)
series((tanh(sinh(x)) - sinh(tanh(x)))/sinh(x^7), x = 0)
series(tanh(x), x = infinity)