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|newpage| Additional Trigonometric functions (real arguments only) =============================================================================== Sine, `x` in degrees, `\mathrm{sind}(x)` ------------------------------------------------------------------------------- .. method:: math53.sind(x) Returns the sine of `x`, with `x` in degrees, `\mathrm{sind}(x)`. See also Wikipedia :cite:p:`WikipediaFun31`, MathWorld :cite:p:`WolframFun31`, NIST :cite:p:`DLMFun30`. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Sind(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Sind('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Sind(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Sind('0.51') Gpr('5.3518479027559984754E-1') Inverse sine, input in degrees, `\mathrm{asind}(x)` ------------------------------------------------------------------------------- .. method:: ctx.asind(x) where ``ctx`` is ``math53``, ``mathc53``, ``ctxcpp``, ``ctxflint``. Returns the inverse sine of `x`, `\mathrm{asin}(x)`. See also Wikipedia :cite:p:`WikipediaFun50`, MathWorld :cite:p:`WolframFun51`, NIST :cite:p:`DLMFun50`, :cite:t:`Ehrhardt2018` (4.2.13), Mpmath :cite:p:`MpmathFun51`. Cosine, `x` in degrees, `\mathrm{cosd}(x)` ------------------------------------------------------------------------------- .. method:: math53.cosd(x) Returns the cosine of `x`, with `x` in degrees, `\mathrm{cosd}(x)`. See also Wikipedia :cite:p:`WikipediaFun30`, MathWorld :cite:p:`WolframFun32`, NIST :cite:p:`DLMFun30`. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Cosd(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Cosd('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Cosd(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Cosd('0.51') Gpr('5.3518479027559984754E-1') Inverse cosine, input in degrees, `\mathrm{acosd}(x)` ------------------------------------------------------------------------------- .. method:: ctx.acosd(x) where ``ctx`` is ``math53``, ``mathc53``, ``ctxcpp``, ``ctxflint``. Returns the inverse cosine of `x`, `\mathrm{acos}(x)`. See also Wikipedia :cite:p:`WikipediaFun50`, MathWorld :cite:p:`WolframFun52`, NIST :cite:p:`DLMFun50`, :cite:t:`Ehrhardt2018` (4.2.3), Flint :cite:p:`FlintFun50`, Flint :cite:p:`FlintFun51`, Mpmath :cite:p:`MpmathFun52`. Tangent, with `x` in degrees, `\mathrm{tand}(x)` ------------------------------------------------------------------------------- .. method:: math53.tand(x) Returns the tangent of `x`, with `x` in degrees, `\mathrm{tand}tan(x)`. See also Wikipedia :cite:p:`WikipediaFun30`, MathWorld :cite:p:`WolframFun33`, NIST :cite:p:`DLMFun30`. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Tand(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Tand('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Tand(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Tand('0.51') Gpr('5.3518479027559984754E-1') Inverse tangent, input in degrees, `\mathrm{atand}(x)` ------------------------------------------------------------------------------- .. method:: ctx.atand(x) where ``ctx`` is ``math53``, ``mathc53``, ``ctxcpp`` or ``ctxflint``. Returns the inverse tangent of `x`, `\mathrm{atan}(x)`. See also Wikipedia :cite:p:`WikipediaFun50`, MathWorld :cite:p:`WolframFun53`, NIST :cite:p:`DLMFun50`, :cite:t:`Ehrhardt2018` (4.2.15), Flint :cite:p:`FlintFun50`, Flint :cite:p:`FlintFun51`, Mpmath :cite:p:`MpmathFun53`. Cotangent, with `x` in degrees, `\mathrm{cotd}(x)` ------------------------------------------------------------------------------- .. method:: math53.cotd(x) Returns the cotangent of `x`, with `x` in degrees, `\mathrm{cotd}(x)`. See also Wikipedia :cite:p:`WikipediaFun30`, MathWorld :cite:p:`WolframFun36`, NIST :cite:p:`DLMFun30`. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Cotd(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Cotd('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Cotd(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Cotd('0.51') Gpr('5.3518479027559984754E-1') Inverse cotangent, input in degrees, `\mathrm{acotd}(x)` ------------------------------------------------------------------------------- .. method:: ctx.acotd(x) Returns the inverse cotangent of `x`, `\mathrm{acot}(x)`. See also Wikipedia :cite:p:`WikipediaFun50`, MathWorld :cite:p:`WolframFun56`, NIST :cite:p:`DLMFun50`, :cite:t:`Ehrhardt2018` (4.2.5). Continuous inverse cotangent, `\mathrm{acotc}(x)` ------------------------------------------------------------------------------- .. method:: ctx.acotc(x) where ``ctx`` is ``math53``, ``mathc53``, ``ctxcpp`` or ``ctxflint``. Returns the continuous inverse cotangent of `x`, `\mathrm{acotc}(x) = \pi/2 - \mathrm{atan}(x)`. See also Wikipedia :cite:p:`WikipediaFun50`, MathWorld :cite:p:`WolframFun56`, NIST :cite:p:`DLMFun50`, Mpmath :cite:p:`MpmathFun56`. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Acotc(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Acotc('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Acotc(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Acotc('0.51') Gpr('5.3518479027559984754E-1') Coversine, `\mathrm{covers}(x) = 1 - \sin(x)` ------------------------------------------------------------------------------- .. method:: math53.covers(x) Returns the `\mathrm{covers}(x) = 1 - \sin(x)`. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Covers(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Covers('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Covers(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Covers('0.51') Gpr('5.3518479027559984754E-1') Versint function `\mathrm{versint}(x) = x - \sin(x)` ------------------------------------------------------------------------------- .. method:: math53.versint (x) Returns `\displaystyle \mathrm{versint}(x) = \int_0^x \mathrm{vers}(t) \mathrm{d}t = x - \sin(x)`, accurate also near 0. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Versint(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Versint('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Versint(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Versint('0.51') Gpr('5.3518479027559984754E-1') Versine function `\mathrm{vers}(x) = 1 - \cos(x)` ------------------------------------------------------------------------------- .. method:: math53.versine(x) Returns the versine function `\mathrm{vers}(x) = 1 - \cos(x)`. See also: https://en.wikipedia.org/wiki/Versine#Haversine An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Versine(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Versine('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Versine(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Versine('0.51') Gpr('5.3518479027559984754E-1') Haversine function `\mathrm{hav}(x) = (1 - \cos(x))/2` ------------------------------------------------------------------------------- .. method:: math53.haversine(x) Returns the haversine function `\mathrm{hav}(x) = (1 - \cos(x))/2 = \sin^2(x/2)`. See also MathWorld :cite:p:`WolframFun302`, Wikipedia :cite:p:`WikipediaFun302`. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Haversine(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Haversine('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Haversine(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Haversine('0.51') Gpr('5.3518479027559984754E-1') Integral of cos powers, `\mathrm{cosint}(n,x)` ------------------------------------------------------------------------------- .. method:: math53.cosint(n,x) Returns `\displaystyle \mathrm{IC}_n(x) = \int_0^x \cos^n(t) \, \mathrm{d}t`, the integral of the nth cos power, for `n \ge 0`. See also :cite:t:`Ehrhardt2018` (3.10.13). An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.CosInt(3, 4) xreal('5.2359877559829887307E-1') >>> xreal.CosInt(3, 12) xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.CosInt(3, 4) Gpr('5.2359877559829887307E-1') >>> Gpr.CosInt(3, 12) Gpr('5.3518479027559984754E-1') Integral of sin powers, `\mathrm{sinint}(n,x)` ------------------------------------------------------------------------------- .. method:: math53.sinint(n,x) Returns sinint(n,x) = integral(sin(t)^n, t=0..x), n >= 0 Returns `\displaystyle \mathrm{IS}_n(x) = \int_0^x \sin^n(t) \, \mathrm{d}t`, the integral of the nth sin power, for `n \ge 0`. See also :cite:t:`Ehrhardt2018` (3.10.14). An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.SinInt(3, 4) xreal('5.2359877559829887307E-1') >>> xreal.SinInt(3, 12) xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.SinInt(3, 4) Gpr('5.2359877559829887307E-1') >>> Gpr.SinInt(3, 12) Gpr('5.3518479027559984754E-1') Solutions of Kepler’s equation, `\mathrm{kepler}(M,e)` ------------------------------------------------------------------------------- .. method:: math53.kepler(M,e) Returns the solutions (eccentric anomaly `x`) of Kepler’s equation from the mean anomaly `M` and the eccentricity `e`, more precisely the solutions `x` of .. math:: M =\begin{cases} x - e \sin(x), & e<1,\\ x+x^3/3, & e=1 \text{ (Barker's equation)},\\ e \sinh(x)-x, & e>1. \end{cases} See also :cite:t:`Ehrhardt2018` (3.10.23). An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Kepler(3, 0.44) xreal('5.2359877559829887307E-1') >>> xreal.Kepler(3, 0.14404) xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Kepler(3, 0.44) Gpr('5.2359877559829887307E-1') >>> Gpr.Kepler(3, 0.14404) Gpr('5.3518479027559984754E-1') Fibonacci function, `F_{\nu}(x)`, of real index `\nu` ------------------------------------------------------------------------------- .. method:: math53.fibfun(v,x) Returns `\displaystyle F_{\nu}(x) = \frac{2^{-\nu}(x+\sqrt{x^2+4})^{\nu} - \cos(\pi\nu) \cdot 2^{\nu} (x+\sqrt{x^2+4})^{-\nu} }{\sqrt{x^2+4}}`, the general Fibonacci function of real index `\nu`. See also :cite:t:`Ehrhardt2018` (3.10.12) , https://en.wikipedia.org/wiki/Fibonacci_polynomials. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Fibfun(3, 4) xreal('5.2359877559829887307E-1') >>> xreal.Fibfun(3, 12) xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Fibfun(3, 4) Gpr('5.2359877559829887307E-1') >>> Gpr.Fibfun(3, 12) Gpr('5.3518479027559984754E-1') Cardinal hyperbolic sine, `\mathrm{sinhc}(x) = \sinh(x)/x` ------------------------------------------------------------------------------- .. method:: math53.sinhc (x) Returns `\mathrm{sinhc}(x) = \sinh(x)/x`, accurate also for `x` near 0. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Sinhc(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Sinhc('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Sinhc(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Sinhc('0.51') Gpr('5.3518479027559984754E-1') Auxiliary function, `\mathrm{sinhmx}(x) = \sinh(x)-x` ------------------------------------------------------------------------------- .. method:: math53.sinhmx (x) Returns sinh(x)-x, accurate also for `x` near 0. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Sinhmx(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Sinhmx('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Sinhmx(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Sinhmx('0.51') Gpr('5.3518479027559984754E-1') Auxiliary function, `\mathrm{coshm1}(x) = \cosh(x)-1` ------------------------------------------------------------------------------- .. method:: math53.coshm1(x) Returns `\cosh(x)-1`, accurate also for `x` near 0. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Coshm1(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Coshm1('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Coshm1(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Coshm1('0.51') Gpr('5.3518479027559984754E-1') Langevin function, `L(x)` ------------------------------------------------------------------------------- .. method:: math53.langevin_l(x) Returns the Langevin function `L(x)`, defined as `L(x) = \coth(x) - 1/x` for `x \ne 0`, and `L(0) = 0` for `x = 0`. See also :cite:t:`Ehrhardt2018` (3.10.16). https://en.wikipedia.org/wiki/Brillouin_and_Langevin_functions An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.LangevinL(0.2) xreal('5.2359877559829887307E-1') >>> xreal.LangevinL(0.21) xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.LangevinL(0.2) Gpr('5.2359877559829887307E-1') >>> Gpr.LangevinL(0.21) Gpr('5.3518479027559984754E-1') Auxiliary function `\log(\cosh(x))` ------------------------------------------------------------------------------- .. method:: math53.logcosh(x) Returns ln(cosh(x)), accurate for x ~ 0 and without overflow for large x An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Logcosh(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Logcosh('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Logcosh(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Logcosh('0.51') Gpr('5.3518479027559984754E-1') Auxiliary function `\log(\sinh(x))` ------------------------------------------------------------------------------- .. method:: math53.logsinh(x) Returns ln(sinh(x)), x > 0, accurate for x ~ 0 and without overflow for large x An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Logsinh(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Logsinh('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Logsinh(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Logsinh('0.51') Gpr('5.3518479027559984754E-1') Auxiliary function, `\mathrm{acos}(1-x)` ------------------------------------------------------------------------------- .. method:: math53.acos1m(x) Returns `\mathrm{acos}(1-x)`, `0 \le x \le 2`, accurate also for `x` near 0. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Acos1m(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Acos1m('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Acos1m(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Acos1m('0.51') Gpr('5.3518479027559984754E-1') Gudermannian function `\mathrm{gd}(x) = \mathrm{asin}(\mathrm{tanh}(x))` ------------------------------------------------------------------------------- .. method:: math53.gd(x) Returns the Gudermannian function `\mathrm{gd}(x) = \mathrm{asin}(\mathrm{tanh}(x))`. See also Wikipedia :cite:p:`WikipediaFun304`, MathWorld :cite:p:`WolframFun304`. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Gudermann(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Gudermann('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Gudermann(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Gudermann('0.51') Gpr('5.3518479027559984754E-1') Inverse haversine function `\mathrm{archav}(x) = \mathrm{acos}(1-2x)` ------------------------------------------------------------------------------- .. method:: math53.archav(z) Returns the inverse haversine function `\mathrm{archav}(x) = \mathrm{acos}(1-2x) 2 \mathrm{asin}(\sqrt{x})`, `0 \le x \le 1`. See also MathWorld :cite:p:`WolframFun303`, Wikipedia :cite:p:`WikipediaFun303`. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.ArcHaversine(0.5) xreal('5.2359877559829887307E-1') >>> xreal.ArcHaversine('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.ArcHaversine(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.ArcHaversine('0.51') Gpr('5.3518479027559984754E-1') Auxiliary function `\mathrm{acosh}(1+x)` ------------------------------------------------------------------------------- .. method:: math53.acosh1p(z) Returns `\mathrm{acosh}(1+x), x \ge 0`, accurate also for `x` near 0. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.Acosh1p(0.5) xreal('5.2359877559829887307E-1') >>> xreal.Acosh1p('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.Acosh1p(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.Acosh1p('0.51') Gpr('5.3518479027559984754E-1') Inverse Gudermannian function `\mathrm{arcgd}(x) = \mathrm{atanh}(\sin(x))` ------------------------------------------------------------------------------- .. method:: math53.arcgd(z) Returns the inverse Gudermannian function `\mathrm{arcgd}(x) = \mathrm{atanh}(\sin(x)), |x| < \pi/2.` See also Wikipedia :cite:p:`WikipediaFun305`, MathWorld :cite:p:`WolframFun305`. An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.ArcGd(0.5) xreal('5.2359877559829887307E-1') >>> xreal.ArcGd('0.51') xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.ArcGd(0.5) Gpr('5.2359877559829887307E-1') >>> Gpr.ArcGd('0.51') Gpr('5.3518479027559984754E-1') Inverse Langevin function, `L^{-1}(x)` ------------------------------------------------------------------------------- .. method:: math53.langevin_l_inv(x) Returns the functional inverse `L^{-1}` of the Langevin function, i.e. `L(L^{-1}(x))= x`, `|x| < 1`. See also :cite:t:`Ehrhardt2018` (3.10.17). https://en.wikipedia.org/wiki/Brillouin_and_Langevin_functions An example in Python .. code-block:: pycon >>> from xlcalcnet import xreal >>> xreal.LangevinLInv(0.2) xreal('5.2359877559829887307E-1') >>> xreal.LangevinLInv(0.21) xreal('5.3518479027559984754E-1') An example in Visual Basic .. code-block:: pycon >>> from xlcalcnet import Gpr >>> Gpr.LangevinLInv(0.2) Gpr('5.2359877559829887307E-1') >>> Gpr.LangevinLInv(0.21) Gpr('5.3518479027559984754E-1')