Wednesday, November 27, 2019

HP Prime: Hyperoblic CAS Transformations

HP Prime:  Hyperbolic CAS Transformations

Introduction

These CAS transforms some expressions involving hyperbolic functions, mainly sinh (hyperbolic sine) and cosh (hyperbolic cosine). 

Let ϕ and Ω be any algebraic expression, real number, or complex number.  These commands are meant to work in CAS mode. 

Exponential Definitions

sinhexp

sinhexp(ϕ) = (e^(ϕ) - e^(-ϕ)) / 2 = ((e^ϕ)^2 - 1) / (2 * e^ϕ)

#cas
sinhexp(f):=
BEGIN
RETURN (e^(f)-e^(−f))/2
END;
#end

coshexp

coshexp(ϕ) =  (e^(ϕ) + e^(-ϕ)) / 2 = ((e^ϕ)^2 + 1) / (2 * e^ϕ)

#cas
coshexp(f):=
BEGIN
RETURN (e^(f)+e^(−f))/2
END;
#end

tanhexp

tanhexp(ϕ) = (e^(ϕ) - e^(-ϕ)) / (e^(ϕ) + e^(-ϕ))

#cas
tanhexp(f):=
BEGIN
RETURN (e^(f)-e^(−f))/
(e^(f)+e^(−f))
END;
#end

Adding Properties

addsinh

addsinh(ϕ + Ω) = sinh ϕ * cosh Ω + sinh Ω * cosh ϕ

#cas
addcosh(f,g):=
BEGIN
RETURN COSH(f)*COSH(g)+
SINH(f)*SINH(g);
END;
#end

addcosh

addcosh(ϕ + Ω) = csoh ϕ * cosh Ω + sinh Ω * sinh ϕ

#cas
addsinh(f,g):=
BEGIN
RETURN SINH(f)*COSH(g)+
COSH(f)*SINH(g);
END;
#end

addtanh

addtanh(ϕ + Ω) = (tanh ϕ + tanh Ω) / (1 + tanh ϕ * tanh Ω)

#cas
addtanh(f,g):=
BEGIN
RETURN (TANH(f)+TANH(g))/
(1+TANH(f)*TANH(g));
END;
#end

Squaring Properties

sqsinh

sqsinh(ϕ) = sinh^2 ϕ = 1/2 * cosh(2 * ϕ) - 1/2

#cas
sqsinh(f):=
BEGIN
RETURN COSH(2*f)/2-1/2;
END;
#end

sqcosh

sqcosh(ϕ) = cosh^2 ϕ = 1/2 * cosh(2 * ϕ) + 1/2

#cas
sqcosh(f):=
BEGIN
RETURN COSH(2*f)/2+1/2;
END;
#end

Product Properties

sinhsinh

sinhsinh(ϕ, Ω) = 1/2 * (cosh(ϕ + Ω) - cosh(ϕ - Ω))

#cas
sinhsinh(f,g):=
BEGIN
RETURN 1/2*(COSH(f+g)-
COSH(f-g));
END;
#end

coshcosh

coshcosh(ϕ, Ω) = 1/2 * (cosh(ϕ + Ω) + cosh(ϕ - Ω))

#cas
coshcosh(f,g):=
BEGIN
RETURN 1/2*(COSH(f+g)+
COSH(f-g));
END;
#end

sinhcosh

sinhcosh(ϕ, Ω) = 1/2 * (sinh(ϕ + Ω) + sinh(ϕ - Ω))

#cas
sinhcosh(f,g):=
BEGIN
RETURN 1/2*(SINH(f+g)+
SINH(f-g));
END;
#end

Source:

Spiegel, Murray R. and Seymour Lipschutz, John Liu.  Schuam's Outlines:  Mathematical Handbook of Formulas and Tables  5th Edition   McGraw Hill: New York  2018  ISBN 978-1-260-01053-4


A little early start to our Thanksgiving feast,

Eddie

All original content copyright, © 2011-2019.  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.

Texas Instruments: TI-36X Pro and TI-30X Pro Mathprint

 Essentially, the Texas Instruments TI-36X Pro and the TI-30X MathPrint are functionally equivalent.  What makes the two calculators differe...