## Special

### integrate function

Returns an approximation for the numerical integral of the given expression over the given range.

#### Syntax

integrate(f,var,a,b)

#### Description

The integrate function returns an approximation for the numerical integral of f with the variable var from a to b. This is mathematically written as:

This integral is the same as the area between the function f and the x-axis from a to b where the area under the axis is counted negative. f may be any function with the variable indicated as the second argument var. a and b may be any numeric expressions that evaluate to real numbers or they can be -INF or INF to indicate negative or positive infinity. integrate does not calculate the integral exactly. Instead the calculation is done using the Gauss-Kronrod 21-point integration rule adaptively to an estimated relative error less than 10-3.

#### Examples

f(x)=integrate(t^2-7t+1, t, -3, 15) will integrate f(t)=t^2-7t+1 from -3 to 15 and evaluate to 396. More useful is f(x)=integrate(s*sin(s), s, 0, x). This will plot the definite integral of f(s)=s*sin(s) from 0 to x, which is the same as the indefinite integral of f(x)=x*sin(x).

#### See also

 Wikipedia MathWorld

### sum function

Returns the summation of an expression evaluated over a range of integers.

sum(f,var,a,b)

#### Description

The sum function returns the summation of f where var is evaluated for all integers from a to b. This is mathematically written as:

f may be any function with the variable indicated as the second argument var. a and b may be any numeric expressions that evaluate to integers.

#### See also

 Wikipedia MathWorld

### product function

Returns the product of an expression evaluated over a range of integers.

#### Syntax

product(f,var,a,b)

#### Description

The product function returns the product of f where var is evaluated for all integers from a to b. This is mathematically written as:

f may be any function with the variable indicated as the second argument var. a and b may be any numeric expressions that evaluate to integers.

#### See also

 Wikipedia MathWorld

### fact function

Returns the factorial of the argument.

fact(n)

#### Description

The fact function returns the factorial of n, commonly written as n!. n may be any numeric expression that evaluates to a positive integer. The function is defined as fact(n)=n(n-1)(n-2)...1, and relates to the gamma function as fact(n)=gamma(n+1).

#### See also

 Wikipedia MathWorld

### gamma function

Returns the value of the Euler gamma function of the argument.

gamma(z)

#### Description

The gamma function returns the result of the Euler gamma function of z, commonly written as Γ(z). z may be any numeric expression that evaluates to a real number or a complex number. The gamma function relates to the factorial function as fact(n)=gamma(n+1). The mathematical definition of the gamma function is:

This cannot be calculated precisely, so Graph is using the Lanczos approximation to calculate the gamma function.

#### See also

 Wikipedia MathWorld

### beta function

Returns the value of the Euler beta function evaluated for the arguments.

beta(m, n)

#### Description

The beta function returns the result of the Euler beta function evaluated for m and n. m and n may be any numeric expressions that evaluate to real numbers or complex numbers. The beta function relates to the gamma function as beta(m, n) = gamma(m) * gamma(n) / gamma(m+n).

#### See also

 Wikipedia MathWorld

### W function

Returns the value of the Lambert W-function evaluated for the argument.

W(z)

#### Description

The W function returns the result of the Lambert W-function, also known as the omega function, evaluated for z. z may be any numeric expression that evaluates to a real number or a complex number. The inverse of the W function is given by f(W)=W*eW.

#### Remarks

For real values of z when z < -1/e, the W function will evaluate to values with an imaginary part.

#### See also

 Wikipedia MathWorld

### zeta function

Returns the value of the Riemann Zeta function evaluated for the argument.

zeta(z)

#### Description

The zeta function returns the result of the Riemann Zeta function, commonly written as ζ(s). z may be any numeric expression that evaluates to a real number or a complex number.

#### Remarks

The zeta function is defined for the whole complex plane except for the pole at z=1.

#### See also

 Wikipedia MathWorld

### mod function

Returns the remainder of the first argument divided by the second argument.

mod(m,n)

#### Description

Calculates m modulo n, the remainder of m/n. mod calculates the remainder f, where m = a*n + f for some integer a. The sign of f is always the same as the sign of n. When n=0, mod returns 0. m and n may be any numeric expressions that evaluate to real numbers.

#### See also

 Wikipedia MathWorld

### dnorm function

Returns the normal distribution of the first argument with optional mean value and standard deviation.

dnorm(x, [μ,σ])

#### Description

The dnorm function is the probability density of the normal distribution, also called Gaussian distribution. x is the variate, also known as the random variable, μ is the mean value and σ is the standard deviation. μ and σ are optional and if left out the standard normal distribution is used where μ=0 and σ=1. x, μ and σ may be any numeric expressions that evaluate to real numbers where σ > 0. The normal distribution is defined as:

#### See also

 Wikipedia MathWorld