abs
(x)¶Return the absolute value of x
, commonly written as |x|. In real mode, it flips the sign of a negative number, thus forcing it to be positive. When given a complex number as argument, it returns the modulus of the number.
The argument can have a dimension.
Example:
abs(-3 meter)
= 3 meter
abs(4 + 3j)
= 5
sqrt
(x)¶Return the square root of x
. If complex numbers are disabled, this function is only defined for x > 0. In complex mode, any complex number may be specified, yielding the complex root in the upper half plane.
The argument may have a dimension.
cbrt
(x)¶Compute the third (cubic) root of x
. In real mode, it accepts any real number. Negative numbers will yield a negative cubic root:
cbrt(-27)
= -3
In complex mode, this function accepts any complex input. The result will generally be the first complex root, i.e. the one with a phase between 0 and π/3. Real negative arguments however will still yield a real (negative) result, matching the function’s behavior in real mode. Use x^(1/3)
to get the first complex root.
exp
(x)¶Compute the natural exponential function.
The argument must be dimensionless.
ln
(x)¶Compute the natural logarithm. In real mode, the argument must be real, with x > 0.
In complex mode, any non-zero number may be given. The result will be the principal value. The branch cut runs across the negative real axis. Nevertheless, in SpeedCrunch ln()
is defined for negative real numbers as ln(-x) = ln(|x|)) + πj, extending the branch from the upper half-plane.
sin
(x)¶Returns the sine of x
. The behavior depends on both the angle unit setting (degrees or radians) and on whether complex numbers are enabled.
In degrees mode, the argument is assumed to be expressed in degrees, such that sin()
is periodic with a period of 360 degrees: sin(x) = sin(x+360). Complex arguments are not allowed in degrees mode, regardless of the corresponding setting.
When radians are set as the angle unit, sin()
will be 2π-periodic. The argument may be complex.
For real arguments beyond approx. |x|>10^{77}, SpeedCrunch no longer recognizes the periodicity of the function and issues an error.
The argument of sin()
must be dimensionless.
The inverse function is arcsin()
.
cos
(x)¶Returns the cosine of x
. The behavior depends on both the angle unit setting (degrees or radians) and on whether complex numbers are enabled.
In degrees mode, the argument is assumed to be expressed in degrees, such that cos()
is periodic with a period of 360 degrees: cos(x) = cos(x+360). Complex arguments are not allowed in degrees mode, regardless of the corresponding setting.
When radians are set as the angle unit, cos()
will be 2π-periodic. The argument may be complex.
For real arguments beyond approx. |x|>10^{77}, SpeedCrunch no longer recognizes the periodicity of the function and issues an error.
The argument of cos()
must be dimensionless.
The inverse function is arccos()
.
tan
(x)¶Returns the tangent of x
. The behavior depends on both the angle unit setting (degrees or radians) and on whether complex numbers are enabled.
In degrees mode, the argument is assumed to be expressed in degrees, such that tan()
is periodic with a period of 180 degrees: tan(x) = tan(x+180). Complex arguments are not allowed in degrees mode, regardless of the corresponding setting.
When radians are set as the angle unit, tan()
will be π-periodic. The argument may be complex.
The argument of tan()
must be dimensionless.
The inverse function is arctan()
.
cot
(x)¶Returns the cotangent of x
. The behavior depends on both the angle unit setting (degrees or radians) and on whether complex numbers are enabled.
In degrees mode, the argument is assumed to be expressed in degrees, such that cot()
is periodic with a period of 180 degrees: cot(x) = cot(x+180). Complex arguments are not allowed in degrees mode, regardless of the corresponding setting.
When radians are set as the angle unit, cot()
will be π-periodic. The argument may be complex.
The argument of cot()
must be dimensionless.
sec
(x)¶Returns the secant of x
, defined as the reciprocal cosine of x
: sec(x) = 1/cos(x). The behavior depends on both the angle unit setting (degrees or radians) and on whether complex numbers are enabled.
In degrees mode, the argument is assumed to be expressed in degrees, such that sec()
is periodic with a period of 360 degrees: sec(x) = sec(x+360). Complex arguments are not allowed in degrees mode, regardless of the corresponding setting.
When radians are set as the angle unit, sec()
will be 2π-periodic. The argument may be complex.
For real arguments beyond approx. |x|>10^{77}, SpeedCrunch no longer recognizes the periodicity of the function and issues an error.
The argument of sec()
must be dimensionless.
csc
(x)¶Returns the cosecant of x
, defined as the reciprocal sine of x
: csc(x) = 1/sin(x). The behavior depends on both the angle unit setting (degrees or radians) and on whether complex numbers are enabled.
In degrees mode, the argument is assumed to be expressed in degrees, such that csc()
is periodic with a period of 360 degrees: csc(x) = csc(x+360). Complex arguments are not allowed in degrees mode, regardless of the corresponding setting.
When radians are set as the angle unit, csc()
will be 2π-periodic. The argument may be complex.
For real arguments beyond approx. |x|>10^{77}, SpeedCrunch no longer recognizes the periodicity of the function and issues an error.
The argument of csc()
must be dimensionless.
arccos
(x)¶Returns the inverse cosine of x
, such that cos(arccos(x)) = x. The behavior of the function depends on both the angle unit setting (degrees or radians) and on whether complex numbers are enabled.
In degrees mode, arccos()
takes a real argument from [-1, 1], and the return value is in the range [0, 180]. Real arguments outside [-1, 1] and complex numbers are not allowed in degrees mode.
When radians are set as the angle unit, arccos()
maps an element from [-1, 1] to a value in [0, π]. When complex numbers are enabled in addition, arccos()
may take any argument from the complex plane. In complex mode, arccos(-1) = π and arccos(1) = 0 will yield the same result as in real mode.
The argument of arccos()
must be dimensionless.
The inverse function is cos()
.
arcsin
(x)¶Returns the inverse sine of x
, such that sin(arcsin(x)) = x. The behavior of the function depends on both the angle unit setting (degrees or radians) and on whether complex numbers are enabled.
In degrees mode, arcsin()
takes a real argument from [-1, 1], and the return value is in the range [-90, 90]. Real arguments outside [-1, 1] and complex numbers are not allowed in degrees mode.
When radians are set as the angle unit, arcsin()
maps an element from [-1, 1] to a value in [-π/2, π/2]. When complex numbers are enabled in addition, arcsin()
may take any argument from the complex plane. In complex mode, arcsin(-1) = π/2 and arcsin(1) = π/2 will yield the same result as in real mode.
The argument of arccos()
must be dimensionless.
The inverse function is sin()
.
arctan
(x)¶Returns the inverse tangent of x
, such that tan(arctan(x)) = x. The behavior of the function depends on both the angle unit setting (degrees or radians) and on whether complex numbers are enabled.
In degrees mode, arctan()
takes a real argument, and the return value is in the range [-90, 90]. Complex arguments are not allowed in degrees mode.
When radians are set as the angle unit, arctan()
maps a real number to a value in [-π/2, π/2]. When complex numbers are enabled in addition, arctan()
may take any argument from the complex plane, except for +j and -j.
The argument of arctan()
must be dimensionless.
The inverse function is tan()
.
arctan2
(x, y)¶Returns the angle formed by the vector (x, y) and the X axis. If the point (x, y) lies in the first quadrant (i.e. both x > 0 and y > 0 are true), it is given by arctan(y/x). However, the function handles vectors in other quadrants as well.
The behavior of the function depends on the angle unit setting (degrees or radians). In degrees mode, this function returns a value in the range ]-180, 180]. When radians are set as the angle unit, the return value lies in the range ]-π, π].
Unlike arctan()
this function only accepts real arguments.
The argument values must be dimensionless.
sinh
(x)¶Return the hyperbolic sine of x
. In complex mode, any complex number may be used as the argument.
The argument must be dimensionless.
The inverse function is arsinh()
.
cosh
(x)¶Return the hyperbolic cosine of x
. In complex mode, any complex number may be used as the argument.
The argument must be dimensionless.
The inverse function is arcosh()
.
tanh
(x)¶Return the hyperbolic tangent of x
. In complex mode, any complex number may be used as the argument.
The argument must be dimensionless.
The inverse function is artanh()
.
arsinh
(x)¶Compute the area hyperbolic sine of x
, the inverse function to sinh()
. arsinh(x) is the only solution to cosh(y) = x.
In complex mode, the function is defined for any complex z
as arsinh(z) = ln[z + (z ^{2} +1) ^{1/2} ].
The function only accepts dimensionless arguments.
arcosh
(x)¶Compute the area hyperbolic cosine of x
, the inverse function to cosh()
. arcosh(x) is the positive solution to cosh(y) = x. Except for x=1, the second solution to this equation will be given by -arcosh(x).
In real mode, the parameter x
must be > 1. In complex mode, the function is defined for any complex z
as arcosh(z) = ln[z + (z ^{2} -1) ^{2} ].
The function only accepts dimensionless arguments.
artanh
(x)¶Compute the area hyperbolic tangent of x
, the inverse function to tanh()
. artanh(x) is the only solution to tanh(y) = x. In real mode, the parameter x
has to fulfill -1 < x < 1.
In complex mode, this function accepts any argument except for -1 and +1. In the complex plane, it is defined as artanh(z) = 1/2 * ln[(z+1)/(z-1)].
The function only accepts dimensionless arguments.
erf
(x)¶Compute the error function, evaluated in x
. The error function is closely related to the Gaussian cumulative density function.
Note that currently only real arguments are allowed. Furthermore, the function only accepts dimensionless arguments.
erfc
(x)¶Compute the complementary error function, evaluated in x
. The complementary error function is defined by erfc(x) = 1 - erf(x)
.
Note that currently only real arguments are allowed. Furthermore, the function only accepts dimensionless arguments.
gamma
(x)¶Evaluates the gamma function (frequently denoted by the Greek letter Γ). The gamma function is an analytic extension to the factorial operation which is defined on real numbers as well. The relation between factorial and the gamma function is given by Γ(n) = (n - 1)!.
Note that currently only real arguments are allowed. Furthermore, the function only accepts dimensionless arguments.
The computation of the factorial operation is in fact implemented via gamma()
. This means that in SpeedCrunch, factorials of non-integer numbers are allowed.
lngamma
(x)¶Computes ln(abs(gamma(x)))
. As the gamma function grows extremely quickly, it is sometimes easier to work with its logarithm instead. lngamma()
allows much larger arguments that would otherwise overflow gamma()
.
Note that currently only real arguments are allowed. Furthermore, the function only accepts dimensionless arguments.
real
(x)¶Return the real part of a complex number x
.
The argument may have a dimension.
imag
(x)¶Return the imaginary part of a complex number x
.
The argument may have a dimension.
phase
(x)¶Returns the phase (angle) of a complex number x
. The unit of the angle corresponds to the current angle mode.
The argument may have a dimension.
See also
abs()
(absolute value)polar
(x)¶Converts the complex number x
to polar form, i.e. the form r e ^{jɸ}. The angle ɸ is always given in radians.
cart
(x)¶Converts the complex number x
to cartesian form, i.e. the form a + j b.
sgn
(x)¶For x >= 0, return +1. For x < 0, return -1.
radians
(x)¶Convert the angle x
into radians. Independently of the angle unit setting, this function will assume that x
is given in degrees and return pi*x/180
.
The function only accepts real, dimensionless arguments.
degrees
(x)¶Convert the angle x
into degrees. Independently of the angle unit setting, this function will assume that x
is given in radians and return 180*x/pi
.
The function only accepts real, dimensionless arguments.
int
(x)¶Returns the integer part of x
, effectively rounding it towards zero.
The function only accepts real, dimensionless arguments.
frac
(x)¶Returns the fractional (non-integer) part of x
, given by frac(x) = x - int(x)
.
The function only accepts real, dimensionless arguments.