convert to partial fraction form - Maple Help

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convert/parfrac - convert to partial fraction form

Calling Sequence

convert(f, parfrac)

convert(f, parfrac, x)

convert(f, parfrac, K)

convert(f, parfrac, x, K)

convert(flist, parfrac, x)

Parameters

f

-

rational function in x

x

-

main variable name

K

-

(optional) real, complex, a field extension, true, false, sqrfree

flist

-

list consisting of the numerator, prefactored denominator, and powers (this is a programmer entry point)

Description

• 

Convert to parfrac performs a partial fraction  decomposition of the rational function f in the variable x.

• 

If no x is provided, parfrac attempts to determine a suitable x, and proceeds if the operation is not ambiguous. For example, an expression that is a rational polynomial in both a and b requires that the variable be specified.

• 

The optional argument K specifies how the denominator in f is to be factored.  If this argument is not specified, the denominator is factored by the factor command, which factors over the field implied by the coefficients present.

• 

If the optional argument K is real (or complex), then a real (complex) floating-point factorization of the denominator is performed.

  

Note: This is implemented only for the univariate case.

• 

If the argument K is RootOf or a radical or a list or set of RootOfs or radicals, then the denominators are factored over the algebraic number field implied by the field extensions K.

• 

If the argument K is the name `sqrfree' then a square-free partial factorization is computed. A square-free factorization of the denominator in x is computed.

• 

If the last argument is `true', this declares that the denominator of f is already in the desired factored form, and no factorization is required.

  

Note: Such a partial fraction decomposition can be done only if the factors in the denominator are relatively prime to each other.

• 

If the programmer entry point form is used, then x must be a name, and the input flist must have the form:

  

 [n,[f1,p1],[f2,p2],...] 

  

where n is the numerator, and f1p1,f2p2,... are the denominator factors. All f1,f2,... must be relatively prime, and all p1,p2,... must be positive integer values.

• 

The programmer form also provides the output in a different form:

  

 [p,[f1,n11,n12,...],[f2,n21,n22,...],...] 

  

where p is the polynomial part, f1,f2,... are as in the input, and the n11,n12,... are the numerators of the partial fraction form such that the algebraic partial fraction form can be obtained as:

  

 p+n11f1+n12f12+...+n21f2+n22f22+... 

• 

Note: The programmer form output is dense, meaning all zero coefficients are included, and the polynomial part is always included (even if zero).

Examples

f:=x5+1x4x2

f:=x5+1x4x2

(1)

convertf,parfrac,x

x+1x11x2

(2)

convertf,parfrac

x+1x11x2

(3)

f:=xxb2

f:=xxb2

(4)

convertf,parfrac,x

bxb2+1xb

(5)

Note: This is an error because Maple cannot determine whether to use x or b.

convertf,parfrac

Error, (in `convert/parfrac`) the variable name (for conversion to partial fractions) must be provided

f:=2.3x5.4x32.3x+1

f:=2.3x5.4x32.3x+1

(6)

convertf,parfrac,x

0.224031228437975x+0.0633606213907298x20.809184744236785x+0.2288540243796240.224031228437975x+0.809184744236785

(7)

convertf,parfrac,x,complex

0.1120156142189870.301653861716895Ix0.4045923721183920.255262681963574I+0.112015614218987+0.301653861716895Ix0.404592372118392+0.255262681963574I0.224031228437975x+0.809184744236785

(8)

f:=4x36x22x42x32x+4

f:=4x36x22x42x32x+4

(9)

convertf,parfrac,x

3x2x32+1x2

(10)

The cubic factor (x^3-2) does not factor over the integers.

convertf,parfrac,x,213

621/3222/3+221/3+4x2+22/3121/321/32x+21/3+2x21/322/3x21/3x2

(11)

convertf,parfrac,x,real

1.00000000000000x1.25992104989487+1.00000000000000x2.+2.x+1.25992104989487x2+1.25992104989487x+1.58740105196820

(12)

convertf,parfrac,x,complex

0.9999999999999994.5202679760439810-16Ix1.25992104989487+1.00000000000000+1.2886888414749810-16Ix+0.629960524947437+1.09112363597172I+1.1.2093129639316210-16Ix+0.6299605249474371.09112363597172I+1.00000000000000+4.4408920985006210-16Ix2.

(13)

p:=x52x42x3+4x2+x2:

f:=36p

f:=36x52x42x3+4x2+x2

(14)

convertf,parfrac,x

9x123x+124x+1+4x2

(15)

convertf,parfrac,x,sqrfree

12x24x212+4x8x21+4x2

(16)

f:=36convertp,sqrfree,x

f:=36x2x212

(17)

convertf,parfrac,x,true

12x24x212+4x8x21+4x2

(18)

Programmer entry point form,

fl:=4x36x22,x2,2,x32,1

fl:=4x36x22,x2,2,x32,1

(19)

pl:=convertfl,parfrac,x

pl:=0,x2,2,1,x32,2x2x2

(20)

which you can compare to the regular form.

f:=fl1muli1i2,i=fl2..1

f:=4x36x22x22x32

(21)

p:=convertf,parfrac,x,true

p:=2x2x2x32+1x22+2x2

(22)

pl1+addaddiji1j1,j=2..nopsi,i=pl2..1

2x2x2x32+1x22+2x2

(23)

See Also

convert/fullparfrac, factor, int, sqrfree


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