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
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
The Psi returns the nth 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
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
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
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
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
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
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
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