Formulas 5. Mathematical functions



abs (x)

absolute value

round (x)

nearest integer; round (1.5) = 2

floor (x)

round down: highest integer value not greater than x

ceiling (x)

round up: lowest integer value not less than x

sqrt (x)

square root: √x, x ≥ 0

min (x, ...)

the minimum of a series of numbers, e.g. min (7.2, 5, 3) = 5

max (x, ...)

the maximum of a series of numbers, e.g. max (7.2, 5, 3) = 7.2

imin (x, ...)

the location of the minimum, e.g. imin (7.2, 5, 3) = 2

imax (x, ...)

the location of the maximum, e.g. imax (7.2, 5, 3) = 1

sin (x)

sine

cos (x)

cosine

tan (x)

tangent

arcsin (x)

arcsine, 1 ≤ x ≤ 1

arccos (x)

arccosine, 1 ≤ x ≤ 1

arctan (x)

arctangent

arctan2 (y, x)

argument angle

sinc (x)

sinus cardinalis: sin (x) / x

sincpi (x)

sinc_{π}: sin (πx) / (πx)

exp (x)

exponentiation: e^{x}; same as e^x

ln (x)

natural logarithm, base e

log10 (x)

logarithm, base 10

log2 (x)

logarithm, base 2

sinh (x)

hyperbolic sine: (e^{x}  e^{x}) / 2

cosh (x)

hyperbolic cosine: (e^{x} + e^{x}) / 2

tanh (x)

hyperbolic tangent: sinh (x) / cosh (x)

arcsinh (x)

inverse hyperbolic sine: ln (x + √(1+x^{2}))

arccosh (x)

inverse hyperbolic cosine: ln (x + √(x^{2}–1))

arctanh (x)

inverse hyperbolic tangent

sigmoid (x)

R → (0,1): 1 / (1 + e^{–x}) or 1 – 1 / (1 + e^{x})

invSigmoid (x)

(0,1) → R: ln (x / (1 – x))

erf (x)

the error function: 2/√π _{0}∫^{x} exp(t^{2}) dt

erfc (x)

the complement of the error function: 1  erf (x)

randomUniform (min, max)

uniform random real number between min (inclusive) and max (exclusive)

randomInteger (min, max)

uniform random integer number between min and max (inclusive)

randomGauss (μ, σ)

Gaussian random real number with mean μ and standard deviation σ

randomPoisson (mean)

Poisson random real number

lnGamma (x)

logarithm of the Γ function

gaussP (z)

the area under the Gaussian distribution between –∞ and z

gaussQ (z)

the area under the Gaussian distribution between z and +∞: the onetailed "statistical significance p" of a value that is z standard deviations away from the mean of a Gaussian distribution

invGaussQ (q)

the value of z for which
gaussQ
(z) = q

chiSquareP (chiSquare, df)

the area under the χ^{2} distribution between 0 and chiSquare, for df degrees of freedom

chiSquareQ (chiSquare, df)

the area under the χ^{2} distribution between chiSquare and +∞, for df degrees of freedom: the "statistical significance p" of the χ^{2} difference between two distributions in df+1 dimensions

invChiSquareQ (q, df)

the value of χ^{2} for which
chiSquareQ
(χ^{2}, df) = q

studentP (t, df)

the area under the student Tdistribution from ∞ to t

studentQ (t, df)

the area under the student Tdistribution from t to +∞

invStudentQ (q, df)

the value of t for which
studentQ
(t, df) = q

fisherP (f, df1, df2)

the area under Fisher's Fdistribution from 0 to f

fisherQ (f, df1, df2)

the area under Fisher's Fdistribution from f to +∞

invFisherQ (q, df1, df2)

the value of f for which
fisherQ
(f, df1, df2) = q

binomialP (p, k, n)

the probability that in n experiments, an event with probability p will occur at most k times

binomialQ (p, k, n)

the probability that in n experiments, an event with probability p will occur at least k times; equals 1 
binomialP
(p, k  1, n)

invBinomialP (P, k, n)

the value of p for which
binomialP
(p, k, n) = P

invBinomialQ (Q, k, n)

the value of p for which
binomialQ
(p, k, n) = Q

hertzToBark (x)

from acoustic frequency to Barkrate (perceptual spectral frequency; place on basilar membrane): 7 ln (x/650 + √(1 + (x/650)^{2}))

barkToHertz (x)

650 sinh (x / 7)

hertzToMel (x)

from acoustic frequency to perceptual pitch: 550 ln (1 + x / 550)

melToHertz (x)

550 (exp (x / 550)  1)

hertzToSemitones (x)

from acoustic frequency to a logarithmic musical scale, relative to 100 Hz: 12 ln (x / 100) / ln 2

semitonesToHertz (x)

100 exp (x ln 2 / 12)

erb (f)

the perceptual equivalent rectangular bandwidth (ERB) in hertz, for a specified acoustic frequency (also in hertz): 6.23·10^{6} f^{2} + 0.09339 f + 28.52

hertzToErb (x)

from acoustic frequency to ERBrate: 11.17 ln ((x + 312) / (x + 14680)) + 43

erbToHertz (x)

(14680 d  312) / (1  d) where d = exp ((x  43) / 11.17)

phonToDifferenceLimens (x)

from perceptual loudness (intensity sensation) level in phon, to the number of intensity difference limens above threshold: 30 · ((61/60)^{ x} – 1).

differenceLimensToPhon (x)

the inverse of the previous: ln (1 + x / 30) / ln (61 / 60).

beta (x, y)

besselI (n, x)

besselK (n, x)
For functions with arrays, see Scripting 5.7. Vectors and matrices.
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© ppgb, July 18, 2017