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


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