In general, any mathematical expression accepted by C, FORTRAN, Pascal, or BASIC is valid. The precedence of these operators is determined by the specifications of the C programming language. White space (spaces and tabs) is ignored inside expressions.

Complex constants are expressed as {<real>,<imag>}, where <real> and <imag> must be numerical constants. For example, {3,2} represents 3 + 2i; {0,1} represents 'i' itself. The curly braces are explicitly required here.

Note that gnuplot uses both "real" and "integer" arithmetic, like FORTRAN and C. Integers are entered as "1", "-10", etc; reals as "1.0", "-10.0", "1e1", 3.5e-1, etc. The most important difference between the two forms is in division: division of integers truncates: 5/2 = 2; division of reals does not: 5.0/2.0 = 2.5. In mixed expressions, integers are "promoted" to reals before evaluation: 5/2e0 = 2.5. The result of division of a negative integer by a positive one may vary among compilers. Try a test like "print -5/2" to determine if your system chooses -2 or -3 as the answer.

The integer expression "1/0" may be used to generate an "undefined" flag,
which causes a point to ignored; the `ternary`

operator gives an example.

The real and imaginary parts of complex expressions are always real, whatever the form in which they are entered: in {3,2} the "3" and "2" are reals, not integers.

The functions in `gnuplot`

are the same as the corresponding functions in
the Unix math library, except that all functions accept integer, real, and
complex arguments, unless otherwise noted.

For those functions that accept or return angles that may be given in either
degrees or radians (sin(x), cos(x), tan(x), asin(x), acos(x), atan(x),
atan2(x) and arg(z)), the unit may be selected by `set angles`

, which
defaults to radians.

The `abs`

function returns the absolute value of its argument. The returned
value is of the same type as the argument.

For complex arguments, abs(x) is defined as the length of x in the complex plane [i.e., sqrt(real(x)**2 + imag(x)**2) ].

The `acos`

function returns the arc cosine (inverse cosine) of its argument.
`acos`

returns its argument in radians or degrees, as selected by `set
angles`.

The `acosh`

function returns the inverse hyperbolic cosine of its argument in
radians.

The `arg`

function returns the phase of a complex number in radians or
degrees, as selected by `set angles`

.

The `asin`

function returns the arc sin (inverse sin) of its argument.
`asin`

returns its argument in radians or degrees, as selected by `set
angles`.

The `asinh`

function returns the inverse hyperbolic sin of its argument in
radians.

The `atan`

function returns the arc tangent (inverse tangent) of its
argument. `atan`

returns its argument in radians or degrees, as selected by
`set angles`

.

The `atan2`

function returns the arc tangent (inverse tangent) of the ratio
of the real parts of its arguments. `atan2`

returns its argument in radians
or degrees, as selected by `set angles`

, in the correct quadrant.

The `atanh`

function returns the inverse hyperbolic tangent of its argument
in radians.

The `besj0`

function returns the j0th Bessel function of its argument.
`besj0`

expects its argument to be in radians.

The `besj1`

function returns the j1st Bessel function of its argument.
`besj1`

expects its argument to be in radians.

The `besy0`

function returns the y0th Bessel function of its argument.
`besy0`

expects its argument to be in radians.

The `besy1`

function returns the y1st Bessel function of its argument.
`besy1`

expects its argument to be in radians.

The `ceil`

function returns the smallest integer that is not less than its
argument. For complex numbers, `ceil`

returns the smallest integer not less
than the real part of its argument.

The `cos`

function returns the cosine of its argument. `cos`

accepts its
argument in radians or degrees, as selected by `set angles`

.

The `cosh`

function returns the hyperbolic cosine of its argument. `cosh`

expects its argument to be in radians.

The `erf`

function returns the error function of the real part of its
argument. If the argument is a complex value, the imaginary component is
ignored.

The `erfc`

function returns 1.0 - the error function of the real part of its
argument. If the argument is a complex value, the imaginary component is
ignored.

The `exp`

function returns the exponential function of its argument (`e`

raised to the power of its argument). On some implementations (notably
suns), exp(-x) returns undefined for very large x. A user-defined function
like safe(x) = x<-100 ? 0 : exp(x) might prove useful in these cases.

The `floor`

function returns the largest integer not greater than its
argument. For complex numbers, `floor`

returns the largest integer not
greater than the real part of its argument.

The `gamma`

function returns the gamma function of the real part of its
argument. For integer n, gamma(n+1) = n!. If the argument is a complex
value, the imaginary component is ignored.

The `ibeta`

function returns the incomplete beta function of the real parts
of its arguments. p, q > 0 and x in [0:1]. If the arguments are complex,
the imaginary components are ignored.

The `inverf`

function returns the inverse error function of the real part
of its argument.

The `igamma`

function returns the incomplete gamma function of the real
parts of its arguments. a > 0 and x >= 0. If the arguments are complex,
the imaginary components are ignored.

The `imag`

function returns the imaginary part of its argument as a real
number.

The `invnorm`

function returns the inverse normal distribution function of
the real part of its argument.

The `int`

function returns the integer part of its argument, truncated
toward zero.

The `lgamma`

function returns the natural logarithm of the gamma function
of the real part of its argument. If the argument is a complex value, the
imaginary component is ignored.

The `log`

function returns the natural logarithm (base `e`

) of its argument.

The `log10`

function returns the logarithm (base 10) of its argument.

The `norm`

function returns the normal distribution function (or Gaussian)
of the real part of its argument.

The `rand`

function returns a pseudo random number in the interval [0:1]
using the real part of its argument as a seed. If seed < 0, the sequence
is (re)initialized. If the argument is a complex value, the imaginary
component is ignored.

The `real`

function returns the real part of its argument.

The `sgn`

function returns 1 if its argument is positive, -1 if its argument
is negative, and 0 if its argument is 0. If the argument is a complex value,
the imaginary component is ignored.

The `sin`

function returns the sine of its argument. `sin`

expects its
argument to be in radians or degrees, as selected by `set angles`

.

The `sinh`

function returns the hyperbolic sine of its argument. `sinh`

expects its argument to be in radians.

The `sqrt`

function returns the square root of its argument.

The `tan`

function returns the tangent of its argument. `tan`

expects
its argument to be in radians or degrees, as selected by `set angles`

.

The `tanh`

function returns the hyperbolic tangent of its argument. `tanh`

expects its argument to be in radians.

A few additional functions are also available.

`column(x)`

may be used only in expressions as part of `using`

manipulations
to fits or datafile plots. See `plot datafile using`

.

The `tm_hour`

function interprets its argument as a time, in seconds from
1 Jan 2000. It returns the hour (an integer in the range 0--23) as a real.

The `tm_mday`

function interprets its argument as a time, in seconds from
1 Jan 2000. It returns the day of the month (an integer in the range 1--31)
as a real.

The `tm_min`

function interprets its argument as a time, in seconds from
1 Jan 2000. It returns the minute (an integer in the range 0--59) as a real.

The `tm_mon`

function interprets its argument as a time, in seconds from
1 Jan 2000. It returns the month (an integer in the range 1--12) as a real.

The `tm_sec`

function interprets its argument as a time, in seconds from
1 Jan 2000. It returns the second (an integer in the range 0--59) as a real.

The `tm_wday`

function interprets its argument as a time, in seconds from
1 Jan 2000. It returns the day of the week (an integer in the range 1--7) as
a real.

The `tm_yday`

function interprets its argument as a time, in seconds from
1 Jan 2000. It returns the day of the year (an integer in the range 1--366)
as a real.

The `tm_year`

function interprets its argument as a time, in seconds from
1 Jan 2000. It returns the year (an integer) as a real.

`valid(x)`

may be used only in expressions as part of `using`

manipulations
to fits or datafile plots. See `plot datafile using`

.

The operators in `gnuplot`

are the same as the corresponding operators in the
C programming language, except that all operators accept integer, real, and
complex arguments, unless otherwise noted. The ** operator (exponentiation)
is supported, as in FORTRAN.

Parentheses may be used to change order of evaluation.

The following is a list of all the unary operators and their usages:

```
Symbol Example Explanation
- -a unary minus
+ +a unary plus (no-operation)
~ ~a * one's complement
! !a * logical negation
! a! * factorial
$ $3 * call arg/column during
````using`

manipulation

(*) Starred explanations indicate that the operator requires an integer argument.

Operator precedence is the same as in Fortran and C. As in those languages, parentheses may be used to change the order of operation. Thus -2**2 = -4, but (-2)**2 = 4.

The factorial operator returns a real number to allow a greater range.

The following is a list of all the binary operators and their usages:

Symbol Example Explanation ** a**b exponentiation * a*b multiplication / a/b division % a%b * modulo + a+b addition - a-b subtraction == a==b equality != a!=b inequality < a<b less than <= a<=b less than or equal to > a>b greater than >= a>=b greater than or equal to & a&b * bitwise AND ^ a^b * bitwise exclusive OR | a|b * bitwise inclusive OR && a&&b * logical AND || a||b * logical OR

(*) Starred explanations indicate that the operator requires integer arguments.

Logical AND (&&) and OR (||) short-circuit the way they do in C. That is,
the second `&&`

operand is not evaluated if the first is false; the second
`||`

operand is not evaluated if the first is true.

There is a single ternary operator:

Symbol Example Explanation ?: a?b:c ternary operation

The ternary operator behaves as it does in C. The first argument (a), which must be an integer, is evaluated. If it is true (non-zero), the second argument (b) is evaluated and returned; otherwise the third argument (c) is evaluated and returned.

The ternary operator is very useful both in constructing piecewise functions and in plotting points only when certain conditions are met.

Examples:

Plot a function that is to equal sin(x) for 0 <= x < 1, 1/x for 1 <= x < 2, and undefined elsewhere:

f(x) = 0<=x && x<1 ? sin(x) : 1<=x && x<2 ? 1/x : 1/0 plot f(x)

Note that `gnuplot`

quietly ignores undefined values, so the final branch of
the function (1/0) will produce no plottable points. Note also that f(x)
will be plotted as a continuous function across the discontinuity if a line
style is used. To plot it discontinuously, create separate functions for the
two pieces. (Parametric functions are also useful for this purpose.)

For data in a file, plot the average of the data in columns 2 and 3 against the datum in column 1, but only if the datum in column 4 is non-negative:

plot 'file' using 1:( $4<0 ? 1/0 : ($2+$3)/2 )

Please see `plot data-file using`

for an explanation of the `using`

syntax.

New user-defined variables and functions of one through five variables may
be declared and used anywhere, including on the `plot`

command itself.

User-defined function syntax:

<func-name>( <dummy1> {,<dummy2>} ... {,<dummy5>} ) = <expression>

where <expression> is defined in terms of <dummy1> through <dummy5>.

User-defined variable syntax:

<variable-name> = <constant-expression>

Examples:

w = 2 q = floor(tan(pi/2 - 0.1)) f(x) = sin(w*x) sinc(x) = sin(pi*x)/(pi*x) delta(t) = (t == 0) ramp(t) = (t > 0) ? t : 0 min(a,b) = (a < b) ? a : b comb(n,k) = n!/(k!*(n-k)!) len3d(x,y,z) = sqrt(x*x+y*y+z*z) plot f(x) = sin(x*a), a = 0.2, f(x), a = 0.4, f(x)

Note that the variable `pi`

is already defined. But it is in no way magic;
you may redefine it to be whatever you like.

Valid names are the same as in most programming languages: they must begin
with a letter, but subsequent characters may be letters, digits, "$", or "_".
Note, however, that the `fit`

mechanism uses several variables with names
that begin "FIT_". It is safest to avoid using such names. "FIT_LIMIT",
however, is one that you may wish to redefine. See the documentation
on `fit`

for details.

See `show functions`

, `show variables`

, and `fit`

.

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