Mathematical Functions

Important Notes

The entries in the table below are the mathematical functions provided in the library. Each of them is a ready-made instance of a libcalculus.Function, which are callable for evaluation but can also be used with methods such as libcalculus.integrate, libcalculus.residue etc., as is documented in Analysis Methods.

Evaluating such a function is very simple - \(\text{sinh}(3)\) can be evaluated using libcalculus.sinh(3). Please note that the variable name in the following table is mostly \(z\) - this is not to imply that those functions can only accept complex inputs; real inputs are accepted as well. Read the Caveats section for more information.

Builtin Functions

Name	Mathematical Notation	Notes
abs	\(z\mapsto\left|z\right|\)	Absolute value
arccos	\(z\mapsto\text{arccos}\left(z\right)\)	Inverse cosine
arccot	\(z\mapsto\text{arccot}\left(z\right)\)	Inverse cotangent
arccsc	\(z\mapsto\text{arccsc}\left(z\right)\)	Inverse cosecant
arcosh	\(z\mapsto\text{arcosh}\left(z\right)\)	Inverse hyperbolic cosine
arcoth	\(z\mapsto\text{arcoth}\left(z\right)\)	Inverse hyperbolic cotangent
arcsch	\(z\mapsto\text{arcsch}\left(z\right)\)	Inverse hyperbolic cosecant
arcsec	\(z\mapsto\text{arcsec}\left(z\right)\)	Inverse secant
arcsin	\(z\mapsto\text{arcsin}\left(z\right)\)	Inverse sine
arctan	\(z\mapsto\text{arctan}\left(z\right)\)	Inverse tangent
arg	\(z\mapsto\text{arg}\left(z\right)\)	Complex argument
arsech	\(z\mapsto\text{arsech}\left(z\right)\)	Inverse hyperbolic secant
arsinh	\(z\mapsto\text{arsinh}\left(z\right)\)	Inverse hyperbolic sine
artanh	\(z\mapsto\text{artanh}\left(z\right)\)	Inverse hyperbolic tangent
conj	\(z\mapsto\bar{z}\)	Complex conjugate
constant(c)	\(z\mapsto c\)	Constant function
cos	\(z\mapsto\text{cos}\left(z\right)\)	Cosine
cosh	\(z\mapsto\text{cosh}\left(z\right)\)	Hyperbolic cosine
cot	\(z\mapsto\text{cot}\left(z\right)\)	Cotangent
coth	\(z\mapsto\text{coth}\left(z\right)\)	Hyperbolic cotangent
csc	\(z\mapsto\text{csc}\left(z\right)\)	Cosecant
csch	\(z\mapsto\text{csch}\left(z\right)\)	Hyperbolic cosecant
e	\(z\mapsto e\)	Euler’s number
exp	\(z\mapsto e^z\)	Exponential function
identity	\(z\mapsto z\)	Identity function
imag	\(z\mapsto\text{Im}\left(z\right)\)	Imaginary part
line(z1, z2)	\(t\mapsto\left(1-t\right)z_1+tz_2\)	Contour representing a line with \(t\in\left[0,1\right]\)
ln	\(z\mapsto\text{ln}\left(z\right)\)	Natural logarithm (principal branch)
pi	\(z\mapsto \pi\)	Constant π
piecewise(comp_, then_, else_)	\(z\mapsto \begin{cases} \text{then_}\left(z \right ) & ; \;\text{comp_}\left(z \right )=\mathbb{T} \\ \text{else_}\left(z \right ) & ; \;\text{comp_}\left(z \right )=\mathbb{F} \end{cases}\)	If comp_ returns True at z return then_(z), otherwise return else_(z).
real	\(z\mapsto\text{Re}\left(z\right)\)	Real part
sec	\(z\mapsto\text{sec}\left(z\right)\)	Secant
sech	\(z\mapsto\text{sech}\left(z\right)\)	Hyperbolic secant
sin	\(z\mapsto\text{sin}\left(z\right)\)	Sine
sinh	\(z\mapsto\text{sinh}\left(z\right)\)	Hyperbolic sine
sphere(center=0, radius=1)	\(t\mapsto \text{center}+\text{radius}\cdot e^{2\pi it}\)	Contour representing a circle with \(t\in\left[0,1\right]\)
tan	\(z\mapsto\text{tan}\left(z\right)\)	Tangent
tanh	\(z\mapsto\text{tanh}\left(z\right)\)	Hyperbolic tangent

Operations

Any standard mathematical operation can be applied to a libcalculus.Function object. Given two libcalculus.Function objects f and g:

Syntax	Mathematical Notation	Notes
-f	\(z\mapsto -f\left(z\right)\)	Function additive inverse
f + g	\(z\mapsto f\left(z\right)+g\left(z\right)\)	Function addition
f - g	\(z\mapsto f\left(z\right)-g\left(z\right)\)	Function subtraction
f * g	\(z\mapsto f\left(z\right)\cdot g\left(z\right)\)	Function multiplication
f / g	\(z\mapsto \frac{f\left(z\right)}{g\left(z\right)}\)	Function division
f ** g	\(z\mapsto \left(f\left(z\right)\right)^{g\left(z\right)}\)	Function exponentiation
f @ g	\(z\mapsto f\left(g\left(z\right)\right)\)	Function composition

The corresponding in-place operations are supported; operations with constants are supported where applicable as well:

>>> (2 * libcalculus.sin)(3)
0.2822400161197344
>>> (4 ** libcalculus.sin)(3)
(1.2160815815872013+0j)

Other Remarks

Try to use library builtins as much as possible - this comes in handy with mathematical constants for example:

f = libcalculus.pi * libcalculus.sin
g = np.pi * libcalculus.sin

In this instance, f.latex() will return "\\pi \\sin\\left(x\\right)" (corresponding to \(\pi \sin\left(x\right)\)), while g.latex() will return "3.14159 \\sin\\left(x\\right)" (corresponding to \(3.14159 \sin\left(x\right)\)).

Caveats

The library employs a fundamental distinction between “purely complex-valued” functions and ones that might occasionally return a complex value: (1j * libcalculus.sin)(3) will call an underlying C++ function accepting a real input and returning a complex output; however, libcalculus.arcsin(3) will actually return np.nan. To avoid this, cast to complex - libcalculus.arcsin(complex(3)) will correctly return (1.5707963267948966+1.762747174039086j).