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Functions and operators

Function

1. A function is a mathematical expression composed of numerical values, parameters, variables, parentheses, and operators.
2. A parameter is specified by continuing a positive integer after character 'p'. The integer must be arranged in order of 0, 1, 2, .... For instance, it becomes p0, p1, p2 and so on.
3. Variables are character 'x' and 'y'.
4. Operators are binary operators, left unary and right unary operators. Refer to the following table.
5. For instance, for a Gaussian function, the mathematical expression becomes p0*exp(-0.5*((x-p1)/p2)^2), where "2" is a numerical value, "p0", "p1" and "p2" are parameters, "x" is a variable, and "^" and "exp()" are operators.
6. If you specify the exponent e(=2.71828) in the function, you can use character 'e'.
7. If you specify (=3.14159...) in the function, you can use character string "pi".

Operators

• Larger the figure in "Priority" in the tables becomes, higher the priority of the operation gets. The largest figure is 6.
• x corresponds to a variable, an arbitrary value or a parameter, a is similar to x and n a positive integer.

Binary operators

Operator	Operation	Priority	Example	Mathematical formula
^	power	6	x^a	xa
*	multiplication	5	a*x	a x
/	division	5	x/a
+	addtion	4	x+a	x + a
-	subtraction	4	x-a	x - a

Left unary operators

Operator	Operation	Priority	Example	Mathematical formula
abs	absolute value	6	abs(x)	|x|
sqr	square	6	sqr(x)	x2
sqrt	square root	6	sqrt(x)
exp	exponential	6	exp(x)	ex
ln	natural logarithm	6	ln(x)	logex
log	common logarithm	6	log(x)	log10x
sin	sine	6	sin(x)	sin x
cos	cosine	6	cos(x)	cos x
tan	tangent	6	tan(x)	tan x
asin	inverse sine	6	asin(x)	sin-1x ( = arcsin x)
acos	inverse cosine	6	acos(x)	cos-1x ( = arccos x)
atan	inverse tangent	6	atan(x)	tan-1x ( = arctan x)
sinh	hyperbolic sine	6	sinh(x)	sinh x
cosh	hyperbolic cosine	6	cosh(x)	cosh x
tanh	hyperbolic tangent	6	tanh(x)	tanh x
asinh	inverse hyperbolic sine	6	asinh(x)	sinh-1x ( = arcsinh x)
acosh	inverse hyperbolic cosine	6	acosh(x)	cosh-1x ( = arccosh x)
atanh	inverse hyperbolic tangent	6	atanh(x)	tanh-1x ( = arctanh x)
cosec	cosecant	6	cosec(x)	cosec x ( = 1/sin x)
sec	secant	6	sec(x)	sec x ( = 1/cos x)
cot	cotangent	6	cot(x)	cot x ( = 1/tan x)
acosec	inverse cosecant	6	acosec(x)	cosec-1 x ( = arccosec x = arcsin 1/x = sin-1 1/x)
asec	inverse secant	6	asec(x)	sec-1 x ( = arcsec x = arccos 1/x = cos-1 1/x)
acot	inverse cotangent	6	acot(x)	cot-1 x ( = arccot x = arctan 1/x = tan-1 1/x)
cosech	hyperbolic cosecant	6	cosech(x)	cosech x (=1/sinh x)
sech	hyperbolic secant	6	sech(x)	sech x (=1/cosh x)
coth	hyperbolic cotangent	6	coth(x)	coth x (=1/tanh x)
acosech	inverse hyperbolic cosecant	6	acosech(x)	cosech-1 x ( = arccosech x = arcsinh 1/x = sinh-11/x)
asech	inverse hyperbolic secant	6	asech(x)	sech-1 x ( = arcsech x = arccosh 1/x = cosh-1 1/x)
acoth	inverse hyperbolic cotangent	6	acoth(x)	coth-1 x ( = arccoth x = arctanh 1/x = tanh-1 1/x)

Right unary operators

Operator	Operation	Priority	Example	Mathematical formula
!	factorial	6	n!
%	percent	6	a%	100 a

Input of parameters

• The values of parameters must be specified in order of p0, p1, p2, ...
• For instance, when an expression "p2*exp(-0.5*((x-p1)/p0)^2)" is specified as a function, the values of the parameters must be inputted in order of "p0", "p1", and "p2".

Examples for functions

Polynomial function

f(x) = p0+p1*x+p2*x^2+....

Exponential function

f(x) = p0*exp(-p1*x)

Gaussian function

f(x) = p0*exp(-0.5*((x-p1)/p2)^2)

p0 : amplitude, p1 : mean(), p2 : standard deviation()

Breit-Wigner(Cauchy-Lorenz) function

f(x) = p0*p1/(p1^2+4*(x-p2)^2)

p0 : amplitude, p1 : Full Width at Half Maximum, p2 : most probable value(peak position)

Landau function

f(x) = p0*exp(-0.5*((x-p1)/p2+exp(-(x-p1)/p2)))

p0 : amplitude, p1 : most probable value(peak position), p2 : width

Ellipse

f(x,y) = p0*x^2+p1*x*y+p2*y^2+p3*x+p4*y-1.0