ZETA

ZETA INDEX

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

The Zeta operator returns Riemann's Zeta function,

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

syntax:

Zeta(<expression>)

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 INDEX

Bernoulli Euler Zeta

BESSELJ INDEX

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

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

syntax:

BesselJ(<order>,<argument>)

examples:


BesselJ(1/2,pi); 

  0 


on rounded; 

BesselJ(0,1); 

  0.765197686558

BESSELY INDEX

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 INDEX

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

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

syntax:

Hankel1(<order>,<argument>)

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 INDEX

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

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

syntax:

Hankel2(<order>,<argument>)

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 INDEX

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

The BesselI operator returns the modified Bessel function I.

syntax:

BesselI(<order>,<argument>)

examples:


on rounded; 

Besseli (1,1); 

  0.565159103992

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

BESSELK INDEX

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

The BesselK operator returns the modified Bessel function K.

syntax:

BesselK(<order>,<argument>)

examples:


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

  - besselk(1,x)

There is currently no numeric support for the operator BesselK .

StruveH INDEX

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

The StruveH operator returns Struve's H function.

syntax:

StruveH(<order>,<argument>)

examples:


struveh(-3/2,x); 

  - besselj(3/2,x) / i

StruveL INDEX

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

The StruveL operator returns the modified Struve L function .

syntax:

StruveL(<order>,<argument>)

examples:


struvel(-3/2,x); 

  besseli(3/2,x)

KummerM INDEX

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

The KummerM operator returns Kummer's M function.

syntax:

KummerM(<parameter>,<parameter>,<argument>)

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 INDEX

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

The KummerU operator returns Kummer's U function.

syntax:

KummerU(<parameter>,<parameter>,<argument>)

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 INDEX

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

The WhittakerW operator returns Whittaker's W function.

syntax:

WhittakerW(<parameter>,<parameter>,<argument>)

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 INDEX

Bessel Functions

Airy_Ai INDEX

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

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

syntax:

Airy_Ai(<argument>)

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 INDEX

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

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

syntax:

Airy_Bi(<argument>)

examples:


Airy_Bi(0); 

  0.614926627446          


Airy_Bi(3.45 + 17.97i); 

  8.80397899932e+9 - 5.5561528511e+9*i

Airy_Aiprime INDEX

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

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

syntax:

Airy_Aiprime(<argument>)

examples:


Airy_Aiprime(0); 

  - 0.258819403793           


Airy_Aiprime(3.45+17.97i);

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

Airy_Biprime INDEX

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

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

syntax:

Airy_Biprime(<argument>)

examples:


Airy_Biprime(0); 


Airy_Biprime(3.45 + 17.97i); 

  3.84251916792e+19 - 2.18006297399e+19*i

Airy Functions INDEX

Airy Functions

Airy_Ai operator

Airy_Bi operator

Airy_Aiprime operator

Airy_Biprime operator

JacobiSN INDEX

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

The Jacobisn operator returns the Jacobi Elliptic function sn.

syntax:

Jacobisn(<expression>,<integer>)

examples:


Jacobisn(0.672, 0.36) 

  0.609519691792 


Jacobisn(1,0.9) 

  0.770085724907881

JacobiCN INDEX

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

The Jacobicn operator returns the Jacobi Elliptic function cn.

syntax:

Jacobicn(<expression>,<integer>)

examples:


Jacobicn(7.2, 0.6) 

  0.837288298482018  


Jacobicn(0.11, 19) 

  0.994403862690043 - 1.6219006985556e-16*i

JacobiDN INDEX

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

The Jacobidn operator returns the Jacobi Elliptic function dn.

syntax:

Jacobidn(<expression>,<integer>)

examples:


Jacobidn(15, 0.683) 

  0.640574162024592 


Jacobidn(0,0) 

  1

JacobiCD INDEX

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

The Jacobicd operator returns the Jacobi Elliptic function cd.

syntax:

Jacobicd(<expression>,<integer>)

examples:


Jacobicd(1, 0.34) 

  0.657683337805273 


Jacobicd(0.8,0.8) 

  0.925587311582301

JacobiSD INDEX

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

The Jacobisd operator returns the Jacobi Elliptic function sd.

syntax:

Jacobisd(<expression>,<integer>)

examples:


Jacobisd(12, 0.4) 

  0.357189729437272    


Jacobisd(0.35,1) 

  - 1.17713873203043

JacobiND INDEX

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

The Jacobind operator returns the Jacobi Elliptic function nd.

syntax:

Jacobind(<expression>,<integer>)

examples:


Jacobind(0.2, 17) 

  1.46553203037507 + 0.0000000000334032759313703*i 


Jacobind(30, 0.001) 

  1.00048958438

JacobiDC INDEX

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

The Jacobidc operator returns the Jacobi Elliptic function dc.

syntax:

Jacobidc(<expression>,<integer>)

examples:


Jacobidc(0.003,1) 

  1 


Jacobidc(2, 0.75) 

  6.43472885111

JacobiNC INDEX

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

The Jacobinc operator returns the Jacobi Elliptic function nc.

syntax:

Jacobinc(<expression>,<integer>)

examples:


Jacobinc(1,0) 

  1.85081571768093 


Jacobinc(56, 0.4387) 

  39.304842663512

JacobiSC INDEX

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

The Jacobisc operator returns the Jacobi Elliptic function sc.

syntax:

Jacobisc(<expression>,<integer>)

examples:


Jacobisc(9, 0.88) 

  - 1.16417697982095  


Jacobisc(0.34, 7) 

  0.305851938390775 - 9.8768100944891e-12*i

JacobiNS INDEX

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

The Jacobins operator returns the Jacobi Elliptic function ns.

syntax:

Jacobins(<expression>,<integer>)

examples:


Jacobins(3, 0.9) 

  1.00945801599785 


Jacobins(0.887, 15) 

  0.683578280513975 - 0.85023411082469*i

JacobiDS INDEX

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

The Jacobisn operator returns the Jacobi Elliptic function ds.

syntax:

Jacobids(<expression>,<integer>)

examples:


Jacobids(98,0.223) 

  - 1.061253961477 


Jacobids(0.36,0.6) 

  2.76693172243692

JacobiCS INDEX

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

The Jacobics operator returns the Jacobi Elliptic function cs.

syntax:

Jacobics(<expression>,<integer>)

examples:


Jacobics(0, 0.767) 

  infinity   


Jacobics(1.43, 0) 

  0.141734127352112

JacobiAMPLITUDE INDEX

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 INDEX

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:

AGM_function(<integer>,<integer>,<integer>)

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 INDEX

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 INDEX

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 INDEX

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

The EllipticK operator returns the Elliptic value K.

syntax:

EllipticK(<integer>)

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 INDEX

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

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

syntax:

EllipticKprime(<integer>)

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 INDEX

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 INDEX

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:

EllipticTheta(<integer>,<expression>,<integer>)

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 INDEX

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

The JacobiZeta operator returns the Jacobian function Zeta.

syntax:

JacobiZeta(<expression>,<integer>)

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 INDEX

Jacobi's Elliptic Functions and Elliptic Integrals

JacobiAMPLITUDE operator

AGM_FUNCTION operator

LANDENTRANS operator

EllipticF operator

EllipticK operator

EllipticKprime operator

EllipticE operator

EllipticTHETA operator

JacobiZETA operator

POCHHAMMER INDEX

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

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

syntax:

Pochhammer(<expression>,<expression>)

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 INDEX

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

The Gamma operator returns the Gamma function.

syntax:

Gamma(<expression>)

examples:


gamma(10); 

  362880    


gamma(1/2); 

  sqrt(pi)

BETA INDEX

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:

Beta(<expression>,<expression>)

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 INDEX

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

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

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

syntax:

Gamma(<expression>)

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 INDEX

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

The Polygamma operator returns the Polygamma function.

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

syntax:

Polygamma(<integer>,<expression>)

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 INDEX

Gamma and Related Functions

DILOG_extended INDEX

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:

Dilog(<order>,<expression>)

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 INDEX

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:

Lambert_W(<z>)

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 INDEX

Miscellaneous Functions

DILOG extended operator

Lambert\_W function operator