Casio Classpad (fx-CP400): Collection of Functions

Casio Classpad  (fx-CP400):  Collection of Functions

For my birthday post, on Pi (π) Day (March 14), here is a collection of functions for the Casio Classpad (300, 330, fx-CP400, fx-CP500). 

Birthday Probability Function:   bday(c,n)

"The probability that a number in a room do not share a same birthday"

c:  number of categories (i.e. days in a year)

n:  population size

Example:

bday(365, 40):  0.1087681902

Percent Change:  pchg(old, new)

old:  old amount

new:  new amount

Example:

pchg(400,500):  25

Combination with Repetition:  nHr(n,r)

n:  population

r:  number of objects to pick

Example:

nHr(52,5):  3819816

Error Function and Error Compliment Function

Error Function:   erf(x)

Error Compliment Function:  erfc(x)

Note that erf(x) + erfc(x) = 1

Example:

erf(0.60):  0.6038560908

erfc(0.60):  0.3961439092

Law of Cosines

a^2 = b^2 + c^2 - 2*b*c*cos(α)

Finding the Angle α:  cosang(a,b,c)

Finding the Side a:  cosside(b,c,α)

Example:

(Degrees Mode)

a = 20.1, b = 18.5, c = 22.3:  cosang(20.1,18.5,22.3):  58.13961834°  (approx)

b = 24.2, c = 18.9, α = 58°:  cosside(24.2,18.9,58):  21.4032954 (approx)

Fresnel Integrals

Fresnel Cosine Integral:  frescos(u)

C(u) = ∫( cos(π * x^2 ÷ 2) dx, 0, u)

Fresnel Sine Integral:  fressin(u)

S(u) = ∫( sin(π * x^2 ÷ 2) dx, 0, u)

Example:

frescos(1.5):  0.445261176

fressin(1.5):  0.6975049601

Bessel Integral of the First Kind:  bessel(n,x)

J_n(x) = 1 / π * ∫( cos(n * t - x * sin t) dt, 0, π)

Example:

bessel(0,1.5):  0.5118276716

bessel(2,1.2):  0.1593490183

Elliptic Integral of the First Kind:  ellip(x)

K(x) = ∫( 1/ √(1 - x^2 * sin^2 t) dt, 0, π/2)

Example:

ellip(-0.6):  1.750753803

ellip(0.4):  1.639999866

Sine Integral:  Si(x)

Si(x) = ∫( sin t / t dt, 0, x)

Example:

Si(1.8):  1.50581678

Si(6):  1.424687551

Beta Function:  beta(a,b)

β(a, b) = (Γ(a) * Γ(b)) ÷ Γ(a+b)

Example:

beta(2,3): 1/12

beta(1.9,4.6):  0.04470413922 (approx)

Relativity Factor:  relat(v)

factor = √(1 - v^2/c^2)

c = 299792458 m/s

v = velocity

Example:

relat(201E6):  0.7419422153  (approx)

Schwarzschild Radius:  schwarz(m)

r = (2 * G * m)/c^2

G = 6.674E-11  m^3/(kg s^2)

c = 299792458 m/s

m = mass the black hole, kg

r = Schwarzschild Radius, m  (event horizon)

Example:

schwarz(7.89E30):  11717.95418

Distance of a Drop:  dropdist(v0,t)

v0:  initial velocity, m/s

t: time, s

Calculated:  distance, m

Example:

dropdist(15,5):  197.583125

Cycle of a Simple Pendulum:  pendu(l)

l:  length of a string, m

Calulated:  time of the pendulum swing, s

Example:  

pendu(5.5):  4.705446883

Impedance in LRC Series Circuit:  lrcser(R,f,L,C)

R:  resistance, Ω

f:  frequency, Hz

L:  inductance, H

C:  capacity, F

Example:

lrcser(4,80,0.1,50E-6):  11.21437564

Impedance in LRC Parallel Circuit:  lrcpar(R,f,L,C)

R:  resistance, Ω

f:  frequency, Hz

L:  inductance, H

C:  capacity, F

Example:

lrcpar(4,80,0.1,50E-6):  3.999122191

Source for:

Distance of a Drop

Cycle of a Simple Pendulum

Impedance in LRC Series Circuit

Impedance in LRC Parallel Circuit

Scientific Calculator 128 fx-1000F/fx-5000F Owner's Manual.   Casio.  Tokyo, Japan. 

Download the file here:  https://drive.google.com/file/d/1M54HlJ9dP95VBEmGzkUozGzJxKGpiHMh/view?usp=share_link

Eddie 

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

HP Prime and TI-84 Plus CE: Digamma Function

HP Prime and TI-84 Plus CE: Digamma Function

Calculation

The HP Prime and TI-84 Plus CE program DIGAM ( ψ  ) calculates the digamma function for any positive integers and multiples of 1/2 (such as 3/2, 5/2, 7/2,etc) using the formulas:

ψ(1/2) = -γ – ln 4 = -1.963510026

ψ(1) = -γ = -0.5772156649

ψ(n) = Σ(1/x, x, 1, n-1) - γ for n is an integer

ψ(n) = Σ(2/(2*x+1), x, 0, n-3/2) - γ – ln 4 for n a multiple of 1/2 for the form p/2, p is odd

where γ = 0.577215664901533 (Euler-Mascheroni constant)

HP Prime Program DIGAM

EXPORT DIGAM(N)

BEGIN

// EWS 2017-12-17

LOCAL K,D,E:=0.577215664901533;

CASE

IF N==1 THEN

RETURN −E; END;

IF N==0.5 THEN

RETURN −E-LN(4); END;

IF FP(N)==0 AND N>1 THEN

RETURN Σ(1/K,K,1,N-1)-E; END;

IF FP(N)==0.5 AND N>1 THEN

RETURN Σ(2/(2*K+1),K,0,N-3/2)-E-LN(4); END;

DEFAULT RETURN "Error: Invalid N";

END;

END;

TI-84 Plus CE Program DIGAM

"DIGAMMA 2017-12-17 EWS"

0.577215664901533→E

0→K

Input "N?",N

If fPart(N)≠0 and fPart(N)≠0.5

Then

Disp "INVALID N"

Stop

End

If N=1

Then

­E→D

End

If fPart(N)=0 and N>1

Then

Σ(1/K,K,1,N-1)-E→D

End

If N=0.5

Then

­E-ln(4)→D

End

If fPart(N)=0.5 and N>1

Then

Σ(2/(2*K+1),K,0,N-1.5)-E-ln(4)→D

End

Disp D

Examples

DIGAM(2) = 0.422784335098

DIGAM(3) = 0.922784335098

DIGAM(4) = 1.25611766843

DIGAM(3/2) = 0.03648997398

DIGAM(5/2) = 0.70315664065

DIGAM(7/2) = 1.10315664065

Source:

Keith Oldham, Jan Myland, & Jerome Spainer.  An Atlas of Functions.  2nd Edition.  Springer: New York.  2009  ISBN 13: 978-0-387-48806-6

Eddie

This blog is property of Edward Shore, 2017

Casio fx-5800p Special Functions

Casio fx-5800p Special Functions

Programs

An Alternate Way of Extracting the Fraction and Integer Parts of a Number

The fraction part is stored in F, and the integer part is stored in I.  This algorithm can be used when a calculator or programming language does not have a fractional part or integer part function.

This program assumes the program is in Radians mode.

Casio fx-5800P Program FPIP

Rad

0.5    → F

? → X

cos(πX) = 0 ⇒ Goto 1

Abs(X) → F

tan^-1 (tan (πF)) ÷ π → F

X < 0 ⇒ -F → F

Lbl 1

F /right-triangle ([Shift] [x^2])

X – F → I

Bernoulli Numbers

The program BERNOULLI approximates the Bernoulli number of nth order.

Casio fx-5800P Program BERNOULLI

? → N

If N = 0

Then

1 → B

Goto 1

IfEnd

If Frac(N ÷ 2) ≠ 0

Then

Int( Abs(N – 2) ÷ (N – 2) – 1) ÷ 4 → B

Goto 1

IfEnd

Σ ( (2*J*π)^(-N), J, 1, 250) * (-1)^(N ÷ 2 + 1) * 2 * N! → B

Lbl 1

B

Euler Numbers

The program EULERNUM calculates the Euler number of order n.

Casio fx-5800P Program EULERNUM

? → N

0 → E

Frac(N ÷ 2) ≠ 0 ⇒ Goto 1

(2 ÷ π)^(N + 1) * N! ÷ 5 → E

-1 – 10 * Int(E) → E

Frac(N ÷ 4) ≠ 0 ⇒ Goto 1

4 – E → E

N ≠ 0 ⇒ Goto 1

1   → E

Lbl 1

E

Custom Formulas

How to create custom formulas:  [MODE], 5.  PROG, 1.  NEW, give the name, 3. Formula

To calculate the custom formula, press [CALC], enter the value for T (ignore X because X is a dummy variable).

Sine Integral:   SI:  ∫( sin(X)/X dX,0,T)

Dawson Integral:  DAWSON:  ∫(e^(T^2-X^2) dX,0,T)

Bessel Function, with order N:  BESSEL:  1/π * ∫(cos(T * sin(X)-N*X) dX,0,T)

Error Function:  ERF:  2/√π * ∫(e^(-X^2) dX,0,T)

Fresnel Sine:  FRESSIN:  ∫( sin(π*X^2/2) dX,0,T)

Fresnel Cosine: FRESCOS: ∫( cos(π*X^2/2) dX,0,T)

Source:

Jerome Spainer and Keith B. Oldham  An Atlas of Functions Hemisphere Publication Corporation: Washington  1987  ISBN 0-89116-5738-8

Eddie

This blog is property of Edward Shore, 2017

HP Prime, Casio Classpad: Bessell Functions of the 1st Kind

Bessel Function of the First Kind

The program BESS1 calculates the Bessel function of the First Kind. With the HP Prime and Casio Classpad, and any calculators that can process integrals (and the processor is fast enough), we can use the definition:

J_n(x) = 1/π * ∫ (cos(n*t – x*sin(t)) dt for t= 0 to t = π

This is the solution to the differential equation:

x^2*y’’ + x*y’ + (x^2 – n^2)*y = 0, where y is a function of x.

There are many other variations and approximations to calculate J_n(x) if the integral function is not available on your calculator. 

More information on the Bessel function is found here:

Wolfram

http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

Wikipedia

https://en.wikipedia.org/wiki/Bessel_function

Casio Classpad (fx-CP400):  The Function BESS1

As the screen above shows, I used a Define statement in the Main Mode:

Define bess1(n,x) = 1/π ∫ (cos(n*t – x*sin(t)) dt, from t = 0 to t = π)

Note:  Make sure that the calculator is in Radians mode before calculating.

HP Prime:  BESS1

EXPORT BESS1(n,t)

BEGIN

// Bessel 1st Kind

LOCAL b;

// Integrate

b:=(1/π)*CAS.int(COS(n*X-t*SIN(X)),X,0,π);

// Approximate

b:=approx(b);

RETURN b;

END;

Note: the integration and approximation are two separate steps.  They need to be in order for this program to work correctly.

Examples

bess1(1,2) ≈ 0.576724807756

bess1(0,6.3) ≈ 0.223812006132

bess1(2,4) ≈ 0.364128145852

This blog is property of Edward Shore, 2016.