CHEBYSHEVT _ _ _ _ _ _ _ _ _ _ _ _ operator

The ChebyshevT operator computes the nth Chebyshev T Polynomial (of the first kind).

syntax:

examples:

ChebyshevT(3,xx); 

          2
  xx*(4*xx   - 3) 



ChebyshevT(3,4); 

  244

Chebyshev's T polynomials are computed using the recurrence relati on:

ChebyshevT(n,x) := 2x*ChebyshevT(n-1,x) - ChebyshevT(n-2,x) with

ChebyshevT(0,x) := 0 and ChebyshevT(1,x) := x

CHEBYSHEVU _ _ _ _ _ _ _ _ _ _ _ _ operator

The ChebyshevU operator returns the nth Chebyshev U Polynomial (of the second kind).

syntax:

examples:

ChebyshevU(3,xx); 

          2
  4*x*(2*x   - 1) 



ChebyshevU(3,4); 

  496

Chebyshev's U polynomials are computed using the recurrence relati on:

ChebyshevU(n,x) := 2x*ChebyshevU(n-1,x) - ChebyshevU(n-2,x) with

ChebyshevU(0,x) := 0 and ChebyshevU(1,x) := 2x

HERMITEP _ _ _ _ _ _ _ _ _ _ _ _ operator

The HermiteP operator returns the nth Hermite Polynomial.

syntax:

examples:

HermiteP(3,xx); 

            2
  4*xx*(2*xx   - 3) 


HermiteP(3,4); 

  464

Hermite polynomials are computed using the recurrence relation:

HermiteP(n,x) := 2x*HermiteP(n-1,x) - 2*(n-1)*HermiteP(n-2,x) with

HermiteP(0,x) := 1 and HermiteP(1,x) := 2x

LAGUERREP _ _ _ _ _ _ _ _ _ _ _ _ operator

The LaguerreP operator computes the nth Laguerre Polynomial. The two argument call of LaguerreP is a (common) abbreviation of LaguerreP(n,0,x).

syntax:

LaguerreP(<integer>,<expression>,<expression>)

examples:

LaguerreP(3,xx); 

       3        2
  (- xx   + 9*xx   - 18*xx + 6)/6



LaguerreP(2,3,4); 

  -2

Laguerre polynomials are computed using the recurrence relation:

LaguerreP(n,a,x) := (2n+a-1-x)/n*LaguerreP(n-1,a,x) - (n+a-1) * LaguerreP(n-2,a,x) with

LaguerreP(0,a,x) := 1 and LaguerreP(2,a,x) := -x+1+a

LEGENDREP _ _ _ _ _ _ _ _ _ _ _ _ operator

The binary LegendreP operator computes the nth Legendre Polynomial which is a special case of the nth Jacobi Polynomial with

LegendreP(n,x) := JacobiP(n,0,0,x)

The ternary form returns the associated Legendre Polynomial (see below).

syntax:

LegendreP(<integer>,<expression>,<expression>)

examples:

LegendreP(3,xx); 

          2
  xx*(5*xx   - 3)
  ----------------
         2



LegendreP(3,2,xx); 

              2
  15*xx*( - xx   + 1)

The ternary form of the operator LegendreP is the associa ted Legendre Polynomial defined as

P(n,m,x) = (-1)**m * (1-x**2)**(m/2) * df(LegendreP(n,x),x,m)

JACOBIP _ _ _ _ _ _ _ _ _ _ _ _ operator

The JacobiP operator computes the nth Jacobi Polynomial.

syntax:

examples:

JacobiP(3,4,5,xx); 

          3         2
  7*(65*xx   - 13*xx   - 13*xx + 1)
  ----------------------------------
                  8



JacobiP(3,4,5,6); 

  94465/8

GEGENBAUERP _ _ _ _ _ _ _ _ _ _ _ _ operator

The GegenbauerP operator computes Gegenbauer's (ultraspherical) polynomials.

syntax:

examples:

GegenbauerP(3,2,xx); 

            2
  4*xx*(8*xx   - 3)



GegenbauerP(3,2,4); 

  2000

SOLIDHARMONICY _ _ _ _ _ _ _ _ _ _ _ _ operator

The SolidHarmonicY operator computes Solid harmonic (Laplace) polynomials.

syntax:

examples:

SolidHarmonicY(3,-2,x,y,z,r2); 

                           2    2
  sqrt(105)*z*(-2*i*x*y + x  - y )
  ---------------------------------
         4*sqrt(pi)*sqrt(2)

SPHERICALHARMONICY _ _ _ _ _ _ _ _ _ _ _ _ operator

The SphericalHarmonicY operator computes Spherical harmonic (Laplace) polynomials. These are special cases of the solid harmonic polynomials, SolidHarmonicY.

syntax:

examples:

SphericalHarmonicY(3,2,theta,phi); 


                                 2          2                               2
  sqrt(105)*cos(theta)*sin(theta) *(cos(phi) +2*cos(phi)*sin(phi)*i-sin(phi) )
  -----------------------------------------------------------------------------
                               4*sqrt(pi)*sqrt(2)

Orthogonal Polynomials

SI _ _ _ _ _ _ _ _ _ _ _ _ operator

The Si operator returns the Sine Integral function.

syntax:

examples:

limit(Si(x),x,infinity); 

  pi / 2 


on rounded; 

Si(0.35); 

  0.347626790989

The numeric values for the operator Si are computed via t he power series representation, which limits the argument range.

SHI _ _ _ _ _ _ _ _ _ _ _ _ operator

The Shi operator returns the hyperbolic Sine Integral function.

syntax:

examples:

df(shi(x),x); 

  sinh(x) / x 


on rounded; 

Shi(0.35); 

  0.352390716351

The numeric values for the operator Shi are computed via the power series representation, which limits the argument range.

S_I _ _ _ _ _ _ _ _ _ _ _ _ operator

The s_i operator returns the Sine Integral function si.

syntax:

examples:

s_i(xx); 

  (2*Si(xx) - pi) / 2 


df(s_i(x),x); 

  sin(x) / x

The operator name s_i is simplified towards SI. Since REDUCE is not case sensitive by default the name ``si'' can't be used.

CI _ _ _ _ _ _ _ _ _ _ _ _ operator

The Ci operator returns the Cosine Integral function.

syntax:

examples:

defint(cos(t)/t,t,x,infinity); 

  - ci (x) 


on rounded; 

Ci(0.35); 

  - 0.50307556932

The numeric values for the operator Ci are computed via t he power series representation, which limits the argument range.

CHI _ _ _ _ _ _ _ _ _ _ _ _ operator

The Chi operator returns the Hyperbolic Cosine Integral function.

syntax:

examples:

defint((cosh(t)-1)/t,t,0,x); 

  - log(x) + psi(1) + chi(x)


on rounded; 

Chi(0.35); 

  - 0.44182471827

The numeric values for the operator Chi are computed via the power series representation, which limits the argument range.

ERF EXTENDED _ _ _ _ _ _ _ _ _ _ _ _ operator

The special function package supplies an extended support for the erf operator which implements the error f unction

defint(e**(-x**2),x,0,infinity) * 2/sqrt(pi)

.

syntax:

examples:

erf(-x); 

  - erf(x)


on rounded; 

erf(0.35); 

  0.379382053562

The numeric values for the operator erf are computed via the power series representation, which limits the argument range.

ERFC _ _ _ _ _ _ _ _ _ _ _ _ operator

The erfc operator returns the complementary Error function

1 - defint(e**(-x**2),x,0,infinity) * 2/sqrt(pi)

.

syntax:

examples:

erfc(xx); 

  - erf(xx) + 1

The operator erfc is simplified towards the erf operator.

EI _ _ _ _ _ _ _ _ _ _ _ _ operator

The Ei operator returns the Exponential Integral function.

syntax:

examples:

df(ei(x),x); 

   x
  e
  ---
  x


on rounded; 

Ei(0.35); 

  - 0.0894340019184

The numeric values for the operator Ei are computed via t he power series representation, which limits the argument range.

FRESNEL_C _ _ _ _ _ _ _ _ _ _ _ _ operator

The Fresnel_C operator represents Fresnel's Cosine function.

syntax:

examples:

int(cos(t^2*pi/2),t,0,x); 

  fresnel_c(x) 


on rounded; 

fresnel_c(2.1); 

  0.581564135061

The operator Fresnel_C has a limited numeric evaluation o f large values of its argument.

FRESNEL_S _ _ _ _ _ _ _ _ _ _ _ _ operator

The Fresnel_S operator represents Fresnel's Sine Integral function.

syntax:

examples:

int(sin(t^2*pi/2),t,0,x); 

  fresnel_s(x) 


on rounded; 

fresnel_s(2.1); 

  0.374273359378

The operator Fresnel_S has a limited numeric evaluation o f large values of its argument.

Integral Functions

BINOMIAL _ _ _ _ _ _ _ _ _ _ _ _ operator

The Binomial operator returns the Binomial coefficient if both parameter are integer and expressions involving the Gamma function otherwise.

syntax:

examples:

Binomial(49,6); 

  13983816 



Binomial(n,3); 

   gamma(n + 1)
  ---------------
  6*gamma(n - 2)

The operator Binomial evaluates the Binomial coefficients from the explicit form and therefore it is not the best algorithm if you want to compute many binomial coefficients with big indices in which case a recursive algorithm is preferable.

STIRLING1 _ _ _ _ _ _ _ _ _ _ _ _ operator

The Stirling1 operator returns the Stirling Numbers S(n,m) of the first kind, i.e. the number of permutations of n symbols which have exactly m cycles (divided by (-1)**(n-m)).

syntax:

examples:

Stirling1 (17,4); 

  -87077748875904 


Stirling1 (n,n-1); 

  -gamma(n+1)
  -------------
  2*gamma(n-1)

The operator Stirling1 evaluates the Stirling numbers of the first kind by rulesets for special cases or by a computing the closed form, which is a series involving the operators BINOMIAL and STIRLING2.

STIRLING2 _ _ _ _ _ _ _ _ _ _ _ _ operator

The Stirling1 operator returns the Stirling Numbers S(n,m) of the second kind, i.e. the number of ways of partitioning a set of n elements into m non-empty subsets.

syntax:

examples:

Stirling2 (17,4); 

  694337290 


Stirling2 (n,n-1); 

   gamma(n+1)
  -------------
  2*gamma(n-1)

The operator Stirling2 evaluates the Stirling numbers of the second kind by rulesets for special cases or by a computing the closed form.

Combinatorial Operators

THREEJSYMBOL _ _ _ _ _ _ _ _ _ _ _ _ operator

The ThreejSymbol operator implements the 3j symbol.

syntax:

examples:

ThreejSymbol({j+1,m},{j+1,-m},{1,0}); 


        j
  ( - 1)  *(abs(j - m + 1) - abs(j + m + 1))
  -------------------------------------------
             3       2                    m
   2*sqrt(2*j   + 9*j   + 13*j + 6)*( - 1)

CLEBSCH_GORDAN _ _ _ _ _ _ _ _ _ _ _ _ operator

The Clebsch_Gordan operator implements the Clebsch_Gordan coefficients. This is closely related to the Threejsymbol.

syntax:

examples:

 Clebsch_Gordan({2,0},{2,0},{2,0}); 


     -2
  ---------
  sqrt(14)

SIXJSYMBOL _ _ _ _ _ _ _ _ _ _ _ _ operator

The SixjSymbol operator implements the 6j symbol.

syntax:

SixjSymbol(<list of j1,j2,j3>,<list of l1,l2,l3>)

examples:

SixjSymbol({7,6,3},{2,4,6}); 

       1
  -------------
  14*sqrt(858)

The operator SixjSymbol uses the ineq package in order to find minima and maxima for the summation index.

3j and 6j symbols

HYPERGEOMETRIC _ _ _ _ _ _ _ _ _ _ _ _ operator

The Hypergeometric operator provides simplifications for the generalized hypergeometric functions. The Hypergeometric operator is included in the package specfn2.

syntax:

examples:

load specfn2;

hypergeometric ({1/2,1},{3/2},-x^2); 


  atan(x)
  --------
     x


hypergeometric ({},{},z); 

   z
  e

The special case where the length of the first list is equal to 2 and the length of the second list is equal to 1 is often called ``the hypergeometric function'' (notated as 2F1(a1,a2,b;x)).

MEIJERG _ _ _ _ _ _ _ _ _ _ _ _ operator

The MeijerG operator provides simplifications for Meijer's G function. The simplifications are performed towards polynomials, elementary or special functions or (generalized) hypergeometric functions.

The MeijerG operator is included in the package specfn2.

syntax:

The first element of the lists has to be the list containing the first group (mostly called ``m'' and ``n'') of parameters. This passes the four parameters of a Meijer's G function implicitly via the length of the lists.

examples:

load specfn2;

MeijerG({{},1},{{0}},x); 

  heaviside(-x+1)


MeijerG({{}},{{1+1/4},1-1/4},(x^2)/4) * sqrt pi;
 


                  2
  sqrt(2)*sin(x)*x
  ------------------
      4*sqrt(x)

Many well-known functions can be written as G functions, e.g. exponentials, logarithms, trigonometric functions, Bessel functions and hypergeometric functions. The formulae can be found e.g. in

A.P.Prudnikov, Yu.A.Brychkov, O.I.Marichev: Integrals and Series, Volume 3: More special functions, Gordon and Breach Science Publishers (1990).

HEAVISIDE _ _ _ _ _ _ _ _ _ _ _ _ operator

The Heaviside operator returns the Heaviside function.

Heaviside(~w) => if (w <0) then 0 else 1

when numberp w;

syntax:

This operator is often included in the result of the simplification of a generalized hypergeometric function or a MeijerG function.

No simplification is done for this function.

ERFI _ _ _ _ _ _ _ _ _ _ _ _ operator

The erfi operator returns the error function of an imaginary argument.

erfi(~x) => 2/sqrt(pi) * defint(e**(t**2),t,0,x);

syntax:

This operator is sometimes included in the result of the simplification of a generalized hypergeometric function or a MeijerG function.

No simplification is done for this function.

Miscellaneous

Special Functions

TAYLOR _ _ _ _ _ _ _ _ _ _ _ _ introduction

This short note describes a package of REDUCE procedures that allow Taylor expansion in one or more variables and efficient manipulation of the resulting Taylor series. Capabilities include basic operations (addition, subtraction, multiplication and division) and also application of certain algebraic and transcendental functions. To a certain extent, Laurent expansion can be performed as well.

TAYLOR _ _ _ _ _ _ _ _ _ _ _ _ operator

The taylor operator is used for expanding an expression into a Taylor series.

syntax:

{,<var>, <expression>,<number>}*)

<expression> can be any valid REDUCE algebraic expression. <var> must be a kernel, and is the expansion variable. The <expression> following it denotes the point about which the expansion is to take place. <number> must be a non-negative integer and denotes the maximum expansion order. If more than one triple is specified taylor will expand its first argument independently with respect to all the variables. Note that once the expansion has been done it is not possible to calculate higher orders.

Instead of a kernel, <var> may also be a list of kernels. In this case expansion will take place in a way so that the sum/ of the degrees of the kernels does not exceed the maximum expansion order. If the expansion point evaluates to the special identifier infinity, taylor tries to expand in a series in 1/<var>.

The expansion is performed variable per variable, i.e. in the example above by first expanding exp(x^2+y^2) with respect to x and then expanding every coefficient with respect to y.

examples:

    taylor(e^(x^2+y^2),x,0,2,y,0,2); 


       2    2    2  2      2  2
  1 + Y  + X  + Y *X  + O(X ,Y )   


    taylor(e^(x^2+y^2),{x,y},0,2); 


       2    2       2  2
  1 + Y  + X  + O({X ,Y })

The following example shows the case of a non-analytical function.

    taylor(x*y/(x+y),x,0,2,y,0,2); 


  ***** Not a unit in argument to QUOTTAYLOR 

Note that it is not generally possible to apply the standard reduce operators to a Taylor kernel. For example, part, coeff, or coeffn cannot be used. Instead, the expression at hand has to be converted to standard form first using the taylortostandard operator.

Differentiation of a Taylor expression is possible. If you differentiate with respect to one of the Taylor variables the order will decrease by one.

Substitution is a bit restricted: Taylor variables can only be replaced by other kernels. There is one exception to this rule: you can always substitute a Taylor variable by an expression that evaluates to a constant. Note that REDUCE will not always be able to determine that an expression is constant: an example is sin(acos(4)).

Only simple taylor kernels can be integrated. More complicated expressions that contain Taylor kernels as parts of themselves are automatically converted into a standard representation by means of the taylortostandard operator. In this case a sui table warning is printed.

TAYLORAUTOCOMBINE _ _ _ _ _ _ _ _ _ _ _ _ switch

If you set taylorautocombine to on, REDUCE automatically combines Taylor expressions during the simplification process. This is equivalent to applying taylorcombine to every expression that contains Taylor kernels. Default is on.

TAYLORAUTOEXPAND _ _ _ _ _ _ _ _ _ _ _ _ switch

taylorautoexpand makes Taylor expressions ``contagious'' in the sense that taylorcombine tries to Taylor expand all non-Taylor subexpressions and to combine the result with the rest. Default is off.

TAYLORCOMBINE _ _ _ _ _ _ _ _ _ _ _ _ operator

This operator tries to combine all Taylor kernels found in its argument into one. Operations currently possible are:

_ _ _ Addition, subtraction, multiplication, and division.

_ _ _ Roots, exponentials, and logarithms.

_ _ _ Trigonometric and hyperbolic functions and their inverses.

examples:

    hugo := taylor(exp(x),x,0,2); 

                  1  2      3
  HUGO := 1 + X + -*X  + O(X )
                  2


    taylorcombine log hugo; 

         3
  X + O(X )


    taylorcombine(hugo + x); 

           1  2      3
  (1 + X + -*X  + O(X )) + X
           2


    on taylorautoexpand; 

    taylorcombine(hugo + x); 

            1  2      3
  1 + 2*X + -*X  + O(X )  
            2

Application of unary operators like log and atan will nearly always succeed. For binary operations their arguments have to be Taylor kernels with the same template. This means that the expansion variable and the expansion point must match. Expansion order is not so important, different order usually means that one of them is truncated before doing the operation.

If taylorkeeporiginal is set to on and if all Taylor kernels in its argument have their original expressions kept taylorcombine will also combine these and store the result as the original expression of the resulting Taylor kernel. There is also the switch taylorautoexpand.

There are a few restrictions to avoid mathematically undefined expressions: it is not possible to take the logarithm of a Taylor kernel which has no terms (i.e. is zero), or to divide by such a beast. There are some provisions made to detect singularities during expansion: poles that arise because the denominator has zeros at the expansion point are detected and properly treated, i.e. the Taylor kernel will start with a negative power. (This is accomplished by expanding numerator and denominator separately and combining the results.) Essential singularities of the known functions (see above) are handled correctly.

TAYLORKEEPORIGINAL _ _ _ _ _ _ _ _ _ _ _ _ switch

taylorkeeporiginal, if set to on, forces the taylor and all Taylor kernel manipulation ope rators to keep the original expression, i.e. the expression that was Taylor expanded. All operations performed on the Taylor kernels are also applied to this expression which can be recovered using the operator taylororiginal. Default is off.

TAYLORORIGINAL _ _ _ _ _ _ _ _ _ _ _ _ operator

Recovers the original expression (the one that was expanded) from the Taylor kernel that is given as its argument.

syntax:

examples:

    hugo := taylor(exp(x),x,0,2); 

                  1  2      3
  HUGO := 1 + X + -*X  + O(X )
                  2


    taylororiginal hugo; 

  ***** Taylor kernel doesn't have an original part in TAYLORORIGINAL


    on taylorkeeporiginal; 

    hugo := taylor(exp(x),x,0,2); 

                  1  2      3
  HUGO := 1 + X + -*X  + O(X )
                  2


    taylororiginal hugo; 

   X
  E   

An error is signalled if the argument is not a Taylor kernel or if the original expression was not kept, i.e. if taylorkeeporiginal was set off durin g expansion.

TAYLORPRINTORDER _ _ _ _ _ _ _ _ _ _ _ _ switch

taylorprintorder, if set to on, causes the remainder to be printed in big-O notation. Otherwise, three dots are printed. Default is on.

TAYLORPRINTTERMS _ _ _ _ _ _ _ _ _ _ _ _ variable

Only a certain number of (non-zero) coefficients are printed. If there are more, an expression of the form n terms is printed to indicate how many non-zero terms have been suppressed. The number of terms printed is given by the value of the shared algebraic variable taylorprintterms. Allowed values are integers and the special identifier all. The latter setting specifies that all terms are to be printed. The default setting is 5.

examples:

    taylor(e^(x^2+y^2),x,0,4,y,0,4); 


       2   1  4    2    2  2                              5  5
  1 + Y  + -*Y  + X  + Y *X  +             (4 terms) + O(X ,Y )
           2


    taylorprintterms := all; 

  TAYLORPRINTTERMS := ALL 


    taylor(e^(x^2+y^2),x,0,4,y,0,4); 


       2   1  4    2    2  2   1  4  2   1  4   1  2  4
  1 + Y  + -*Y  + X  + Y *X  + -*Y *X  + -*X  + -*Y *X
           2                   2         2      2
     1  4  4      5  5
   + -*Y *X  + O(X ,Y )
     4
      

TAYLORREVERT _ _ _ _ _ _ _ _ _ _ _ _ operator

taylorrevert allows reversion of a Taylor series of a function f, i.e., to compute the first terms of the expansion of the inverse of f from the expansion of f.

syntax:

The first argument must evaluate to a Taylor kernel with the second argument being one of its expansion variables.

examples:

    taylor(u - u**2,u,0,5); 

       2      6
  U - U  + O(U ) 


    taylorrevert (ws,u,x); 

       2      3      4       5      6
  X + X  + 2*X  + 5*X  + 14*X  + O(X )  

TAYLORSERIESP _ _ _ _ _ _ _ _ _ _ _ _ operator

This operator may be used to determine if its argument is a Taylor kernel.

syntax:

examples:

    hugo := taylor(exp(x),x,0,2); 

                  1  2      3
  HUGO := 1 + X + -*X  + O(X )
                  2


    if taylorseriesp hugo then OK;

  OK 


    if taylorseriesp(hugo + y) then OK else NO; 


  NO  

Note that this operator is subject to the same restrictions as, e.g., ordp or numberp, i.e. it may only be used in boolean expressions in if or let statements.

TAYLORTEMPLATE _ _ _ _ _ _ _ _ _ _ _ _ operator

The template of a Taylor kernel, i.e. the list of all variables with respect to which expansion took place together with expansion point and order can be extracted using

syntax:

This returns a list of lists with the three elements (VAR,VAR0,ORDER). An error is signalled if the argument is not a Taylor kernel.

examples:

    hugo := taylor(exp(x),x,0,2); 

                  1  2      3
  HUGO := 1 + X + -*X  + O(X )
                  2


    taylortemplate hugo; 

  {{X,0,2}}  

TAYLORTOSTANDARD _ _ _ _ _ _ _ _ _ _ _ _ operator

This operator converts all Taylor kernels in its argument into standard form and resimplifies the result.

syntax:

examples:

    hugo := taylor(exp(x),x,0,2); 

                  1  2      3
  HUGO := 1 + X + -*X  + O(X )
                  2


    taylortostandard hugo; 

   2
  X  + 2*X + 2
  ------------  
       2

Taylor series

GNUPLOT AND REDUCE _ _ _ _ _ _ _ _ _ _ _ _ introduction

The GNUPLOT system provides easy to use graphics output for curves or surfaces which are defined by formulas and/or data sets. GNUPLOT supports a great variety of output devices such as X-windows, VGA screen, postscript, picTeX. The REDUCE GNUPLOT package lets one use the GNUPLOT graphical output directly from inside REDUCE, either for the interactive display of curves/surfaces or for the production of pictures on paper.

Note that this package may not be supported on all system platforms.

For a detailed description you should read the GNUPLOT system documentation, available together with the GNUPLOT installation material from several servers by anonymous FTP.

The REDUCE developers thank the GNUPLOT people for their permission to distribute GNUPLOT together with REDUCE.