hyp2exp
09-01-2022, 12:18 PM
Post: #1
 robmio Member Posts: 172 Joined: Jan 2020
hyp2exp
Hello, is there the "inverse" function of "hyp2exp"? That is, is there a function that does the reverse process of "hyp2exp"?

Best regards, Roberto.
09-04-2022, 08:00 PM
Post: #2
 parisse Senior Member Posts: 1,326 Joined: Dec 2013
RE: hyp2exp
No, but it's fairly easy: subst(expression,exp,cosh+sinh)
09-05-2022, 07:21 AM
Post: #3
 robmio Member Posts: 172 Joined: Jan 2020
RE: hyp2exp
(09-04-2022 08:00 PM)parisse Wrote:  No, but it's fairly easy: subst(expression,exp,cosh+sinh)

Thank you very much, parisse

Another question: why are inverse hyperbolic functions (ATANH, ACOSH, etc.) automatically represented by the corresponding logarithmic form?

Sincerely, robmio
09-05-2022, 11:50 AM
Post: #4
 Albert Chan Senior Member Posts: 2,697 Joined: Jul 2018
RE: hyp2exp
(09-04-2022 08:00 PM)parisse Wrote:  No, but it's fairly easy: subst(expression,exp,cosh+sinh)

Above exp2hyp expression might be "simplified" to sin/cos
Just keep in mind, e^(i*x) = cos(x) + i*sin(x) = cosh(i*x) + sinh(i*x)

cos(x) = cosh(i*x)      i*sin(x) = sinh(i*x)
cosh(x) = cos(i*x)      i*sinh(x) = sin(i*x)

CAS> subst(e^(i*x), exp, x -> cosh(x) + sinh(x))

cos(x) + i*sin(x)

---

Same trick can be used for a "better" exp2trig

CAS> f := e^x - e^-x                                   // = 2*sinh(x)
CAS> exp2trig(f)      → e^x - e^-x
CAS> f(exp = (x -> cos(i*x) + sin(i*x)/i))     → -2*i*sin(i*x)

Or, in steps, since sinh/cosh tends to "simplify" to sin/cos

CAS> f(exp=cosh+sinh)(x=x/i)(x=x*i)          → -2*i*sin(i*x)
09-05-2022, 12:40 PM
Post: #5
 parisse Senior Member Posts: 1,326 Joined: Dec 2013
RE: hyp2exp
(09-05-2022 07:21 AM)robmio Wrote:  Another question: why are inverse hyperbolic functions (ATANH, ACOSH, etc.) automatically represented by the corresponding logarithmic form?

Sincerely, robmio
For a CAS, it is always simpler to work with a smaller set of functions. cosh and sinh are still used sometimes (in the French curriculum system), they are not replaced, atanh and the like are never used, hence they are replaced (same as for sec/csc etc.)
09-05-2022, 04:25 PM
Post: #6
 Albert Chan Senior Member Posts: 2,697 Joined: Jul 2018
RE: hyp2exp
Is this a bug?
acosh(z) should produce non-negative real part, same as acos(z)

CAS> acosh(-2.) // ???

1.31695789692+3.14159265359*i

CAS> acosh(-2+1e-15i)

1.31695789692+3.14159265359*i

CAS> acosh(-2-1e-15i)

1.31695789692-3.14159265359*i
09-09-2022, 11:46 AM
Post: #7
 robmio Member Posts: 172 Joined: Jan 2020
RE: hyp2exp
(09-05-2022 11:50 AM)Albert Chan Wrote:
(09-04-2022 08:00 PM)parisse Wrote:  No, but it's fairly easy: subst(expression,exp,cosh+sinh)

Above exp2hyp expression might be "simplified" to sin/cos
Just keep in mind, e^(i*x) = cos(x) + i*sin(x) = cosh(i*x) + sinh(i*x)

cos(x) = cosh(i*x)      i*sin(x) = sinh(i*x)
cosh(x) = cos(i*x)      i*sinh(x) = sin(i*x)

CAS> subst(e^(i*x), exp, x -> cosh(x) + sinh(x))

cos(x) + i*sin(x)

---

Same trick can be used for a "better" exp2trig

CAS> f := e^x - e^-x                                   // = 2*sinh(x)
CAS> exp2trig(f)      → e^x - e^-x
CAS> f(exp = (x -> cos(i*x) + sin(i*x)/i))     → -2*i*sin(i*x)

Or, in steps, since sinh/cosh tends to "simplify" to sin/cos

CAS> f(exp=cosh+sinh)(x=x/i)(x=x*i)          → -2*i*sin(i*x)

Hello, is there an algorithm that translates the logarithmic formulas of the inverse hyperbolic functions into the inverse hyperbolic functions represented with "acosh", "asinh", "atanh", etc.?
For instance:

e^x*ln(x+√(x^2+1))-ln(ln(√(x+1)*√(x-1)+x)) --instruction--> exp(x)*asinh(x)-ln(acosh(x))

Best regards, robmio.
09-13-2022, 04:00 PM
Post: #8
 robmio Member Posts: 172 Joined: Jan 2020
RE: hyp2exp
(09-05-2022 12:40 PM)parisse Wrote:
(09-05-2022 07:21 AM)robmio Wrote:  Another question: why are inverse hyperbolic functions (ATANH, ACOSH, etc.) automatically represented by the corresponding logarithmic form?

Sincerely, robmio
For a CAS, it is always simpler to work with a smaller set of functions. cosh and sinh are still used sometimes (in the French curriculum system), they are not replaced, atanh and the like are never used, hence they are replaced (same as for sec/csc etc.)

Dear professor, is there the "inverse" function of "atrig2ln"? That is, is there a function that does the reverse process of "atrig2ln"?

Best regards, Roberto.
09-14-2022, 05:48 PM
Post: #9
 parisse Senior Member Posts: 1,326 Joined: Dec 2013
RE: hyp2exp
evalc converts complex ln to inverse trig functions. However for Inverse hyperbolic functions, you will have to implement it yourself.
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