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product

definite and indefinite product

Product

inert form of product

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

product(f,k)

kf

product(f,k=m..n)

k=mnf

product(f,k=alpha)

k=αf

product(f,k=expr)

k=exprf

Product(f,k)

kf

Product(f,k=m..n)

k=mnf

Product(f,k=alpha)

k=αf

Product(f,k=expr)

k=exprf

Parameters

f

-

expression

k

-

name, the product index

m, n

-

integers or arbitrary expressions

alpha

-

RootOf

expr

-

expression not containing k

Description

• 

The product command is for computing symbolic products. It is used to compute a formula for an indefinite or definite product. If a formula cannot be computed, Maple returns the product unevaluated. A typical example would be product(x+k, k=0..n-1) which returns the formula GAMMA(x+n)/GAMMA(x). If you want to multiply a finite sequence of values, rather than compute a formula, use the mul command.  For example mul(x+k, k=0..2) returns x*(x+1)*(x+2). Although the product command can be used to compute explicit products, the mul command should be used in programs for explicit products.

• 

You can enter the product command using either the 1-D or 2-D calling sequence.  For example, product(x+k, k=0..n-1) is equivalent to k=0n1x+k.

• 

The call product(f, k) computes the indefinite product of f(k) with respect to k. That is, it computes a formula g such that g(k+1)/g(k) = f(k) for all k.

• 

The call product(f, k = m..n) computes the definite product of f(k) over the given range m..n, that is, it computes f(m) f(m+1) ... f(n). The definite product is equivalent to g(n+1)/g(m) where g is the indefinite product. For example, product(n,n) = product(k, k=1..n-1) = GAMMA(n).

• 

If m = n+1 then the value returned is 1. If m > n+1 then 1/product(f, k=n+1..m-1) is the value returned.

• 

The call product(f, k= alpha) computes the definite product of f(k) over the roots of a polynomial where alpha must be a RootOf.

• 

The call product(f, k= expr) substitutes the value of expr for k in f.

• 

If Maple cannot find a closed form for the product, the function call itself is returned. (The prettyprinter displays the product function using a stylized product sign.)

• 

The capitalized function name Product is the inert product function, which simply returns unevaluated.  It appears gray so that it is easily distinguished from a returned product calling sequence.

Examples

k=14k2

576

(1)

For a finite sequence of values, use the mul command.

mulk2,k=1..4

576

(2)

k=1nk2

Γn+12

(3)

kk2

Γk2

(4)

k=04ak

a0a1a2a3a4

(5)

k=0nak

k=0nak

(6)

k=0mn+k=k=0mn+k

k=0mn+k=Γn+m+1Γn

(7)

k=15kx

120x5

(8)

k=x5xk

Γ5x+1Γx

(9)

k=1∞1k

0

(10)

Compatibility

• 

The product command was updated in Maple 2016; see Advanced Math.

See Also

mul

resultant

RootOf

sum

 


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