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==== Math-Special Commands ====

=== Beta ===

Calculates the beta function.    Beta(x,y) = (x-1)! * (y-1)! / (x+y-1)!  

** Syntax **

** Input: **

** Home - Algebraic Entry **
** Home - Textbook Entry**
** Program Editor **

CAS.Beta(number, number)

** Home - RPN Entry**

2:  number
1:  number
CAS.Beta(2),  press Enter

** CAS Mode **

Beta(number or algebraic expression, number or algebraic expression)

Numbers can be real or complex.

** Output: **

Beta(x,y) = (x-1)! * (y-1)! / (x+y-1)!

** Examples: **

Beta(8,5) returns 1/3960

Beta(1+7*i, -2*i) returns .00118622003099-.0037324363508*i

** Access: **  Toolbox, Math, 8. Special, 1. Beta

** See Also:  ** Gamma, Psi
 
=== Gamma ===

Calculates the gamma function.    Gamma(x) = integral of t^(x-1) * e^-t dt from 0 to infinity

Also Gamma(x) = (x-1)! 

** Syntax **

** Input: **

** Home - Algebraic Entry **
** Home - Textbook Entry **
** Program Editor **

CAS.Gamma(number)

** Home - RPN Entry **

1:  number
CAS.Gamma(1),  press Enter

** CAS Mode **

Gamma(number)

** Output : ** Result of the Gamma function. See the formulas above.

** Notes:  **  Gamma for complex numbers work best in CAS mode.  Factorial and gamma of complex numbers in Home mode cannot be approximated.

** Examples: **

Gamma(5) returns 24
Gamma(-8) returns Inf in Home Mode, ±∞ in CAS Mode

** Related: ** Psi, Beta 

=== Psi  ===

The Psi returns the //n//th derivative of the digamma function.  The digamma function is defined as:

Ψ(x) = d/dz ln Γ(x)

The general syntax of Psi is Psi(number, order).   If //order// is omitted, then digamma function is calculated.

** Syntax **

** Input: **

** Home Mode - Algebraic Entry **
** Home Mode - Textbook Entry **
** Program Editor **

CAS.Psi(number, order)

** Home - RPN Entry **

2:  number
1:  order
CAS.Psi(2), press Enter

For the digamma function:
1:  number
CAS.Psi(1), press Enter

** CAS Mode **

Psi(number, order)

** Output: **   Result of the polygamma function.  See the formulas above.

** Examples: **

Digamma:   
Psi(6) returns -euler_gamma + 137/60, approximately 1.70611766843

Polygamma:
Psi(6,2) returns -2 *(Zeta(3) - 256103/216000), approximately -.032789732245

** Notes: **  euler_gamma is the gamma constant, which is approximately 0.577215664902.

** Access: **  Toolbox, Math, 8. Special, 3. Psi

** See Also:  ** Gamma, Psi, Beta


=== Zeta ===

Calculates the Zeta (ζ)  function.  The general formula for the Zeta function is:

ζ(x) = Σ (k^-x) from k = 1 to ∞

** Syntax **

Zeta(real number).  

Caution:  Complex numbers return a result with the Eta function, which does not give full approximate answers.  (rev. 5166) 

** Input: **

** Home Mode - Algebraic Entry **
** Home Mode - Textbook Entry **
** Program Editor **

CAS.Zeta(real number)

** Home Mode - RPN Entry **

1:  real number
CAS.Zeta(1), press Enter

** CAS Mode **

Zeta(real number)

** Output: **  Result of the zeta function.  If the number is an even integer, an exact answer is returned.  
If the number is an odd integer, Zeta(odd integer) is returned, which can be approximated.  Otherwise, an approximate answer is returned.

** Examples: **

Zeta(3) returns Zeta(3),which is approximately 1.20205690316
Zeta(6) returns π^6/945, which is approximately 1.01734306198
Zeta(3.25) returns 1.15915198568

** Access: **  Toolbox, Math, 8. Special, 4.  Zeta

** See Also:  ** Gamma
 
=== erf ===

Calculates the error (erf)  function.  The general formula for the error function is:

erf(x) = 2/√π * ∫ e^(-t^2) dt from 0 to x

** Syntax **

erf(number)

The number can either be a real or complex number.  

** Input: **

** Home Mode - Algebraic Entry **
** Home Mode - Textbook Entry **
** Program Editor **

CAS.erf(number)

** Home Mode - RPN Entry **

1:  number
CAS.erf(1), press Enter

** CAS Mode **

erf(number)

** Output: **  Result of the error function.  

Home Mode:  erf(number) is returned.

CAS Mode Only:  If the number is an integer, erf(integer) is returned, which can be approximated.  If the number is not an integer, an approximation is returned.

** Examples: **

CAS Mode:
approx(erf(2)) returns .995322265019
approx(erf(3)) returns .999977909503

** Access: **  Toolbox, Math, 8. Special, 5. erf

** See Also:  ** erfc
 
=== erfc ===

Calculates the error compliment (erfc)  function.  The general formula for the error compliment function is:

erfc(x) = 1 - erf(x) 

** Syntax **

erf(number)

The number can either be a real or complex number.  

** Input: **

** Home Mode - Algebraic Entry **
** Home Mode - Textbook Entry **
** Program Editor **

CAS.erfc(number)

** Home Mode - RPN Entry **

1:  number
CAS.erfc(1), press Enter

** CAS Mode **

erfc(number)

** Output: **  Result of the error complement function.  

Home Mode:  1 - erf(number) is returned.

CAS Mode Only:  If the number is an integer, 1 - erf(integer) is returned, which can be approximated.  If the number is not an integer, an approximation is returned.

** Examples: **

CAS Mode:
approx(erfc(2)) returns .00467773498105
approx(erfc(3)) returns 2.20904969837E-5

** Access: **  Toolbox, Math, 8. Special, 6. erfc

** See Also:  ** erf
 
===  Ei  ===

Calculates the exponential integral:  Ei(x) = ∫ e^-t/t dt from x to ∞

** Syntax **

Ei(number)

The number can either be a real or complex number.  

** Input: **

** Home Mode - Algebraic Entry **
** Home Mode - Textbook Entry **
** Program Editor **

CAS.Ei(number)

** Home Mode - RPN Entry **

1:  number
CAS.Ei(1), press Enter

** CAS Mode **

Ei(number)

** Output: **  Result of the exponential function.  

CAS Mode, Home Mode with Textbook or Algebraic Entry:  If the number is an integer, then Ei(integer) or CAS.Ei(integer), respectively, is returned, which can be approximated.  If the number is not an integer, or a complex number, then an approximation is returned.

Home Mode with RPN Entry:  Ei(number) is returned if the number is an integer.  Otherwise, an approximate answer is returned.

** Examples: **

CAS Mode:
approx(Ei(2)) returns 4.954234356
approx(Ei(3)) returns 9.93383257063

** Access: **  Toolbox, Math, 8. Special, 7.  Ei

** See Also:  ** Si, Ci

===  Si  ===

Calculates the sine  integral:  Si(x) = ∫ sin t/t dt from 0 to t

** Syntax **

Si(number)

The number must be a real number.  A "Bad Argument Value" error occurs when the argument is a complex number.  

** Input: **

** Home Mode - Algebraic Entry **
** Home Mode - Textbook Entry **
** Program Editor **

CAS.Si(number)

** Home Mode - RPN Entry **

1:  number
CAS.Si(1), press Enter

** CAS Mode **

Si(number)

** Output: **  Result of the sine integral function.  

CAS Mode, Home Mode with Textbook or Algebraic Entry:  If the number is an integer, then Si(integer) or CAS.Si(integer), respectively, is returned, which can be approximated.  If the number is not an integer, or a complex number, then an approximation is returned.

Home Mode with RPN Entry:  Si(number) is returned.

** Examples: **

CAS Mode:
approx(Si(2)) returns1.6054129768

** Access: **  Toolbox, Math, 8. Special, 8. Si

** See Also:  ** Ei, Ci
 
===  Ci  ===

Calculates the cosine  integral:  Ci(x) = euler_gamma + ln x + ∫ (cos t - t)/t dt from 0 to t
where euler_gamma is the gamma constant, which is approximately 0.577215664902.

** Syntax **

Ci(number)

The number must be a real number.  A "Bad Argument Value" error occurs when the argument is a complex number.  

** Input: **

** Home Mode - Algebraic Entry **
** Home Mode - Textbook Entry **
** Program Editor **

CAS.Ci(number)

** Home Mode - RPN Entry **

1:  number
CAS.Ci(1), press Enter

** CAS Mode **

Ci(number)

** Output: **  Result of the sine integral function.  

CAS Mode, Home Mode with Textbook or Algebraic Entry:  If the number is an integer, then Ci(integer) or CAS.Ci(integer), respectively, is returned, which can be approximated.  If the number is not an integer, or a complex number, then an approximation is returned.

Home Mode with RPN Entry:  Ci(number) is returned.

** Examples: **

CAS Mode:
approx(Ci(2)) returns  .422980828775

** Access: **  Toolbox, Math, 8. Special,9.  Ci

** Source: ** http://en.wikipedia.org/wiki/Trigonometric_integral#Cosine_integral

** See Also:  ** Ei, Si

Return to the Prime Programming Page:  [[Prime:Programming|Prime Programming Language (PPL)]]
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