ZETA _ _ _ _ _ _ _ _ _ _ _ _ operator

The Zeta operator returns Riemann's Zeta function,

Zeta (z) := sum(1/(k**z),k,1,infinity)

syntax:

examples:

Zeta(2); 

    2
  pi  / 6 


on rounded; 

Zeta 1.01; 

  100.577943338

Numerical computation for the Zeta function for arguments close to 1 are tedious, because the series is converging very slowly. In this case a formula (e.g. found in Bender/Orzag: Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill) is used.

No numerical approximation for complex arguments is done.

Bernoulli Euler Zeta

BESSELJ _ _ _ _ _ _ _ _ _ _ _ _ operator

The BesselJ operator returns the Bessel function of the first kind.

syntax:

examples:

BesselJ(1/2,pi); 

  0 


on rounded; 

BesselJ(0,1); 

  0.765197686558  

BESSELY _ _ _ _ _ _ _ _ _ _ _ _ operator

The BesselY operator returns the Bessel function of the second kind.

syntax:

BesselY(<order>,<argument>)

examples:

BesselY (1/2,pi); 

  - sqrt(2) / pi 


on rounded; 

BesselY (1,3); 

  0.324674424792

The operator BesselY is also called Weber's function.

HANKEL1 _ _ _ _ _ _ _ _ _ _ _ _ operator

The Hankel1 operator returns the Hankel function of the first kind.

syntax:

examples:

on complex; 

Hankel1 (1/2,pi); 

  - i * sqrt(2) / pi 


Hankel1 (1,pi); 

  besselj(1,pi) + i*bessely(1,pi)

The operator Hankel1 is also called Bessel function of th e third kind. There is currently no numeric evaluation of Hankel functions.

HANKEL2 _ _ _ _ _ _ _ _ _ _ _ _ operator

The Hankel2 operator returns the Hankel function of the second kind.

syntax:

examples:

on complex; 

Hankel2 (1/2,pi); 

  - i * sqrt(2) / pi 


Hankel2 (1,pi); 

  besselj(1,pi) - i*bessely(1,pi)

The operator Hankel2 is also called Bessel function of th e third kind. There is currently no numeric evaluation of Hankel functions.

BESSELI _ _ _ _ _ _ _ _ _ _ _ _ operator

The BesselI operator returns the modified Bessel function I.

syntax:

examples:

on rounded; 

Besseli (1,1); 

  0.565159103992

The knowledge about the operator BesselI is currently fai rly limited.

BESSELK _ _ _ _ _ _ _ _ _ _ _ _ operator

The BesselK operator returns the modified Bessel function K.

syntax:

examples:

df(besselk(0,x),x); 

  - besselk(1,x)

There is currently no numeric support for the operator BesselK .

STRUVEH _ _ _ _ _ _ _ _ _ _ _ _ operator

The StruveH operator returns Struve's H function.

syntax:

examples:

struveh(-3/2,x); 

  - besselj(3/2,x) / i

STRUVEL _ _ _ _ _ _ _ _ _ _ _ _ operator

The StruveL operator returns the modified Struve L function .

syntax:

examples:

struvel(-3/2,x); 

  besseli(3/2,x)

KUMMERM _ _ _ _ _ _ _ _ _ _ _ _ operator

The KummerM operator returns Kummer's M function.

syntax:

examples:

kummerm(1,1,x); 

   x
  e  


on rounded; 

kummerm(1,3,1.3); 

  1.62046942914

Kummer's M function is one of the Confluent Hypergeometric functio ns. For reference see the hypergeometric operator.

KUMMERU _ _ _ _ _ _ _ _ _ _ _ _ operator

The KummerU operator returns Kummer's U function.

syntax:

examples:

df(kummeru(1,1,x),x) 

  - kummeru(2,2,x)

Kummer's U function is one of the Confluent Hypergeometric functio ns. For reference see the hypergeometric operator.

WHITTAKERW _ _ _ _ _ _ _ _ _ _ _ _ operator

The WhittakerW operator returns Whittaker's W function.

syntax:

examples:

WhittakerW(2,2,2); 

                    1
  4*sqrt(2)*kummeru(-,5,2)
                    2
  -------------------------
             e

Whittaker's W function is one of the Confluent Hypergeometric func tions. For reference see the hypergeometric operator.

Bessel Functions

AIRY_AI _ _ _ _ _ _ _ _ _ _ _ _ operator

The Airy_Ai operator returns the Airy Ai function for a given argument.

syntax:

examples:

on complex;
on rounded;
Airy_Ai(0); 


  0.355028053888          


Airy_Ai(3.45 + 17.97i); 

  - 5.5561528511e+9 - 8.80397899932e+9*i  

AIRY_BI _ _ _ _ _ _ _ _ _ _ _ _ operator

The Airy_Bi operator returns the Airy Bi function for a given argument.

syntax:

examples:

Airy_Bi(0); 

  0.614926627446          


Airy_Bi(3.45 + 17.97i); 

  8.80397899932e+9 - 5.5561528511e+9*i   

AIRY_AIPRIME _ _ _ _ _ _ _ _ _ _ _ _ operator

The Airy_Aiprime operator returns the Airy Aiprime function for a given argument.

syntax:

examples:

Airy_Aiprime(0); 

  - 0.258819403793           


Airy_Aiprime(3.45+17.97i);

  - 3.83386421824e+19 + 2.16608828136e+19*i 

AIRY_BIPRIME _ _ _ _ _ _ _ _ _ _ _ _ operator

The Airy_Biprime operator returns the Airy Biprime function for a given argument.

syntax:

examples:

Airy_Biprime(0); 


Airy_Biprime(3.45 + 17.97i); 

  3.84251916792e+19 - 2.18006297399e+19*i

Airy Functions

JACOBISN _ _ _ _ _ _ _ _ _ _ _ _ operator

The Jacobisn operator returns the Jacobi Elliptic function sn.

syntax:

examples:

Jacobisn(0.672, 0.36) 

  0.609519691792 


Jacobisn(1,0.9) 

  0.770085724907881 

JACOBICN _ _ _ _ _ _ _ _ _ _ _ _ operator

The Jacobicn operator returns the Jacobi Elliptic function cn.

syntax:

examples:

Jacobicn(7.2, 0.6) 

  0.837288298482018  


Jacobicn(0.11, 19) 

  0.994403862690043 - 1.6219006985556e-16*i  

JACOBIDN _ _ _ _ _ _ _ _ _ _ _ _ operator

The Jacobidn operator returns the Jacobi Elliptic function dn.

syntax:

examples:

Jacobidn(15, 0.683) 

  0.640574162024592 


Jacobidn(0,0) 

  1 

JACOBICD _ _ _ _ _ _ _ _ _ _ _ _ operator

The Jacobicd operator returns the Jacobi Elliptic function cd.

syntax:

examples:

Jacobicd(1, 0.34) 

  0.657683337805273 


Jacobicd(0.8,0.8) 

  0.925587311582301 

JACOBISD _ _ _ _ _ _ _ _ _ _ _ _ operator

The Jacobisd operator returns the Jacobi Elliptic function sd.

syntax:

examples:

Jacobisd(12, 0.4) 

  0.357189729437272    


Jacobisd(0.35,1) 

  - 1.17713873203043  

JACOBIND _ _ _ _ _ _ _ _ _ _ _ _ operator

The Jacobind operator returns the Jacobi Elliptic function nd.

syntax:

examples:

Jacobind(0.2, 17) 

  1.46553203037507 + 0.0000000000334032759313703*i 


Jacobind(30, 0.001) 

  1.00048958438  

JACOBIDC _ _ _ _ _ _ _ _ _ _ _ _ operator

The Jacobidc operator returns the Jacobi Elliptic function dc.

syntax:

examples:

Jacobidc(0.003,1) 

  1 


Jacobidc(2, 0.75) 

  6.43472885111  

JACOBINC _ _ _ _ _ _ _ _ _ _ _ _ operator

The Jacobinc operator returns the Jacobi Elliptic function nc.

syntax:

examples:

Jacobinc(1,0) 

  1.85081571768093 


Jacobinc(56, 0.4387) 

  39.304842663512  

JACOBISC _ _ _ _ _ _ _ _ _ _ _ _ operator

The Jacobisc operator returns the Jacobi Elliptic function sc.

syntax:

examples:

Jacobisc(9, 0.88) 

  - 1.16417697982095  


Jacobisc(0.34, 7) 

  0.305851938390775 - 9.8768100944891e-12*i 

JACOBINS _ _ _ _ _ _ _ _ _ _ _ _ operator

The Jacobins operator returns the Jacobi Elliptic function ns.

syntax:

examples:

Jacobins(3, 0.9) 

  1.00945801599785 


Jacobins(0.887, 15) 

  0.683578280513975 - 0.85023411082469*i 

JACOBIDS _ _ _ _ _ _ _ _ _ _ _ _ operator

The Jacobisn operator returns the Jacobi Elliptic function ds.

syntax:

examples:

Jacobids(98,0.223) 

  - 1.061253961477 


Jacobids(0.36,0.6) 

  2.76693172243692 

JACOBICS _ _ _ _ _ _ _ _ _ _ _ _ operator

The Jacobics operator returns the Jacobi Elliptic function cs.

syntax:

examples:

Jacobics(0, 0.767) 

  infinity   


Jacobics(1.43, 0) 

  0.141734127352112 

JACOBIAMPLITUDE _ _ _ _ _ _ _ _ _ _ _ _ operator

The JacobiAmplitude operator returns the amplitude of u.

syntax:

JacobiAmplitude(<expression>,<integer>)

examples:

JacobiAmplitude(7.239, 0.427) 

  0.0520978301448978 


JacobiAmplitude(0,0.1) 

  0 

Amplitude u = asin(Jacobisn(u,m))

AGM_FUNCTION _ _ _ _ _ _ _ _ _ _ _ _ operator

The AGM_function operator returns a list of (N, AGM, list of aNtoa0, list of bNtob0, list of cNtoc0) where a0, b0 and c0 are the initial values; N is the index number of the last term used to generate the AGM. AGM is the Arithmetic Geometric Mean.

syntax:

examples:

AGM_function(1,1,1) 

  1,1,1,1,1,1,0,1  


AGM_function(1, 0.1, 1.3) 

  {6,
   2.27985615996629, 
   {2.27985615996629, 2.27985615996629,
    2.2798561599706, 2.2798624278857, 
    2.28742283656583, 2.55, 1},
   {2.27985615996629, 2.27985615996629,
    2.27985615996198, 2.2798498920555, 
    2.27230201920557, 2.02484567313166, 4.1},
   {0, 4.30803136219904e-12, 0.0000062679151007581,
    0.00756040868012758, 0.262577163434171, - 1.55, 5.9}}

The other Jacobi functions use this function with initial values a0=1, b0=sqrt(1-m), c0=sqrt(m).

LANDENTRANS _ _ _ _ _ _ _ _ _ _ _ _ operator

The landentrans operator generates the descending landen transformation of the given imput values, returning a list of these values; initial to final in each case.

syntax:

landentrans(<expression>,<integer>)

examples:

landentrans(0,0.1) 

  {{0,0,0,0,0},{0.1,0.0025041751943776, 


 

  0.00000156772498954046,6.1444078 9914461e-13,0}}  

The first list ascends in value, and the second descends in value.

ELLIPTICF _ _ _ _ _ _ _ _ _ _ _ _ operator

The EllipticF operator returns the Elliptic Integral of the First Kind.

syntax:

EllitpicF(<expression>,<integer>)

examples:

EllipticF(0.3, 8.222) 

  0.3 


EllipticF(7.396, 0.1) 

  7.58123216114307 

The Complete Elliptic Integral of the First Kind can be found by putting the first argument to pi/2 or by using EllipticK and the second argument.

ELLIPTICK _ _ _ _ _ _ _ _ _ _ _ _ operator

The EllipticK operator returns the Elliptic value K.

syntax:

examples:

EllipticK(0.2) 

  1.65962359861053   


EllipticK(4.3) 

  0.808442364282734 - 1.05562492399206*i  


EllipticK(0.000481) 

  1.57098526617635    

The EllipticK function is the Complete Elliptic Integral of the First Kind.

ELLIPTICKPRIME _ _ _ _ _ _ _ _ _ _ _ _ operator

The EllipticK' operator returns the Elliptic value K(m).

syntax:

examples:

EllipticKprime(0.2) 

  2.25720532682085 


EllipticKprime(4.3) 

  1.05562492399206 


EllipticKprime(0.000481) 

  5.206621921966   

The EllipticKprime function is the Complete Elliptic Inte gral of the First Kind of (1-m).

ELLIPTICE _ _ _ _ _ _ _ _ _ _ _ _ operator

The EllipticE operator used with two arguments returns the Elliptic Integral of the Second Kind.

syntax:

EllipticE(<expression>,<integer>)

examples:

EllipticE(1.2,0.22) 

  1.15094019180949 


EllipticE(0,4.35) 

  0                


EllipticE(9,0.00719) 

  8.98312465929145  

The Complete Elliptic Integral of the Second Kind can be obtained by using just the second argument, or by using pi/2 as the first argument.

The EllipticE operator used with one argument returns the Elliptic value E.

syntax:

EllipticE(<integer>)

examples:

EllipticE(0.22) 

  1.48046637439519  


EllipticE(pi/2, 0.22) 

  1.48046637439519  

ELLIPTICTHETA _ _ _ _ _ _ _ _ _ _ _ _ operator

The EllipticTheta operator returns one of the four Theta functions. It cannot except any number other than 1,2,3 or 4 as its first argument.

syntax:

examples:

EllipticTheta(1, 1.4, 0.72) 

  0.91634775373  


EllipticTheta(2, 3.9, 6.1 ) 

  -48.0202736969 + 20.9881034377 i 


EllipticTheta(3, 0.67, 0.2) 

  1.0083077448   


EllipticTheta(4, 8, 0.75) 

  0.894963369304 


EllipticTheta(5, 1, 0.1) 

  ***** In EllipticTheta(a,u,m); a = 1,2,3 or 4.   

Theta functions are important because every one of the Jacobian Elliptic functions can be expressed as the ratio of two theta functions.

JACOBIZETA _ _ _ _ _ _ _ _ _ _ _ _ operator

The JacobiZeta operator returns the Jacobian function Zeta.

syntax:

examples:

JacobiZeta(3.2, 0.8) 

  - 0.254536403439 


JacobiZeta(0.2, 1.6) 

  0.171766095970451 - 0.0717028569800147*i  

The Jacobian function Zeta is related to the Jacobian function The ta. But it is significantly different from Riemann's Zeta Function Zeta.

Jacobi's Elliptic Functions and Elliptic Integrals

POCHHAMMER _ _ _ _ _ _ _ _ _ _ _ _ operator

The Pochhammer operator implements the Pochhammer notation (shifted factorial).

syntax:

examples:

pochhammer(17,4); 

  116280 



pochhammer(1/2,z); 

    factorial(2*z)
  --------------------
    2*z
  (2   *factorial(z))

A number of complex rules for Pochhammer are inactive, be cause they cause a huge system load in algebraic mode. If one wants to use more rules for the simplification of Pochhammer's notation, one can do:

let special!*pochhammer!*rules;

GAMMA _ _ _ _ _ _ _ _ _ _ _ _ operator

The Gamma operator returns the Gamma function.

syntax:

examples:

gamma(10); 

  362880    


gamma(1/2); 

  sqrt(pi)

BETA _ _ _ _ _ _ _ _ _ _ _ _ operator

The Beta operator returns the Beta function defined by

Beta (z,w) := defint(t**(z-1)* (1 - t)**(w-1),t,0,1) .

syntax:

examples:

Beta(2,2); 

  1 / 6 


Beta(x,y); 

  gamma(x)*gamma(y) / gamma(x + y)

The operator Beta is simplified towards the GAMMA operator.

PSI _ _ _ _ _ _ _ _ _ _ _ _ operator

The Psi operator returns the Psi (or DiGamma) function.

Psi(x) := df(Gamma(z),z)/ Gamma (z)

syntax:

examples:

Psi(3); 

  (2*log(2) + psi(1/2) + psi(1) + 3)/2 


on rounded; 

- Psi(1); 

  0.577215664902

Euler's constant can be found as - Psi(1).

POLYGAMMA _ _ _ _ _ _ _ _ _ _ _ _ operator

The Polygamma operator returns the Polygamma function.

Polygamma(n,x) := df(Psi(z),z,n);

syntax:

examples:

 Polygamma(1,2); 

     2
  (pi   - 6) / 6


on rounded; 

Polygamma(1,2.35); 

  0.52849689109

The Polygamma function is used for simplification of the ZETA function for some arguments.

Gamma and Related Functions

DILOG EXTENDED _ _ _ _ _ _ _ _ _ _ _ _ operator

The package specfn supplies an extended support for the dilog operator which implements the dilog arithm function.

dilog(x) := - defint(log(t)/(t - 1),t,1,x);

syntax:

examples:

defint(log(t)/(t - 1),t,1,x); 

  - dilog (x) 


dilog 2; 

      2
  - pi  /12 



on rounded; 

Dilog 20; 

  - 5.92783972438

The operator Dilog is sometimes called Spence's Integral for n = 2.

LAMBERT\_W FUNCTION _ _ _ _ _ _ _ _ _ _ _ _ operator

Lambert's W function is the inverse of the function w * e**w. It is used in the solve package for equations containing exponentials and logarithms.

syntax:

examples:

Lambert_W(-1/e); 

  -1 


solve(w + log(w),w); 

  w=lambert_w(1)


on rounded; 

Lambert_W(-0.05); 

  - 0.0527059835515

The current implementation will compute the principal branch in rounded mode only.

Miscellaneous Functions