Custom Function

Your own equation in x with up to five parameters, A to E. Use it when the theory of your experiment gives a form that no built-in model has, and fit that form directly instead of linearizing it.

Fit the example data Start with your own data

Example: NIST’s Misra1a adsorption data, 14 points, opened as a new copy you can edit.

Parameters

: First parameter. Use the letters in order: A, then B, and so on.
: Second parameter.
: Third parameter.
: Fourth parameter.
: Fifth parameter. Write powers of ten with a lowercase e, such as 1.2e3, so they are not read as E.

Tips

Write the equation in Python syntax: * for multiplication, ^ or ** for powers, and parentheses for grouping. Pasted −, × and ÷ are accepted. The equation is checked as you type, and a matching built-in model is suggested when there is one.
Custom equations have no automatic starting values: a blank Guess starts at 2. Enter a guess for every parameter, ideally within a factor of a few of the answer. Fix a parameter at a known value, or give it limits, to keep it in a physical range.
Trigonometric functions use radians. For an angle in degrees, write radians(x). Logarithms need positive arguments, and sqrt and fractional powers non-negative ones, over every x and the range where you evaluate the model.
Custom equations use numerical derivatives, so a built-in model with the same form can converge faster. Saved fits and reports show the equation typeset, and the guide explains how to choose starting values.

Functions and constants

Write	Meaning
`sin(x)`, `cos(x)`, `tan(x)`	Sine, cosine and tangent of an angle in radians
`asin(x)`, `acos(x)`, `atan(x)`	Inverse sine, cosine and tangent, in radians
`sinh(x)`, `cosh(x)`, `tanh(x)`, `coth(x)`	Hyperbolic sine, cosine, tangent and cotangent
`asinh(x)`, `acosh(x)`, `atanh(x)`, `acoth(x)`	Inverse hyperbolic sine, cosine, tangent and cotangent
`exp(x)`	e raised to the power x. Euler’s number itself is `exp(1)`
`log(x)`, `ln(x)`	Natural logarithm, base e. For base b, write `log(x)/log(b)`
`log10(x)`	Logarithm base 10
`sqrt(x)`	Square root
`abs(x)`	Absolute value
`floor(x)`, `ceil(x)`	Round down or up to a whole number
`sinc(x)`	Normalized sinc, sin(πx)/(πx). For sin(u)/u, write `sinc(u/pi)`
`erf(x)`, `erfc(x)`	Error function and complementary error function, 1 − erf(x)
`fact(x)`	Factorial x!, computed as Γ(x + 1), so non-integer x is accepted
`degrees(x)`, `radians(x)`	Convert radians to degrees, or degrees to radians
`besselj0(x)`, `besselj1(x)`	Bessel functions of the first kind, J₀ and J₁, for circular apertures and membranes
`pi`, `π`	The constant π = 3.14159…

Examples

A*exp(-x/B)*cos(2*pi*x/C + D) + E: Damped oscillation with an offset, fitting the period C directly
A*exp(-(x-B)^2/(2*C^2)) + D*x + E: Gaussian peak on a sloping background, also built in as Gaussian on Linear Background
A/sqrt(1 + (x/B)^2): Low-pass filter gain against frequency; B is the corner frequency
A*cos(radians(x) - B)^2 + C: Malus’s law with the analyzer angle in degrees
A*B^x*exp(-B)/fact(x): Poisson counts with mean B, the same as the built-in Poisson model
A*tanh((x - B)/C) + D: Smooth step between two levels, centered at B with width C
A*(2*besselj1(B*x)/(B*x))^2 + C: Circular-aperture (Airy) diffraction pattern against angle; leave out x = 0, where it is 0/0

Related models

Guide: Fitting a custom equation