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De la misma forma que las razones trigonométricas seno y coseno son funciones de una variable real con la propiedad
sen
2
(x)+
cos
2
(x)=1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4CaiaabwgacaqGUbWaaWbaaSqabeaacaaIYaaaaOGaaiikaiaadIhacaGGPaGaey4kaSIaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaaiikaiaadIhacaGGPaGaeyypa0JaaGymaaaa@44C4@
podemos construir un par de funciones reales f y g tales que el punto
(f(t),g(t))
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadAgacaGGOaGaamiDaiaacMcacaGGSaGaam4zaiaacIcacaWG0bGaaiykaiaacMcaaaa@3E71@
recorra la hipérbola
x
2
-
y
2
=1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4CaiaabwgacaqGUbWaaWbaaSqabeaacaaIYaaaaOGaaiikaiaadIhacaGGPaGaey4kaSIaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaaiikaiaadIhacaGGPaGaeyypa0JaaGymaaaa@44C4@
Se trata de las funciones hiperbólicas seno y coseno; en símbolos, sinh(x) y cosh(x). Sin entrar en un análisis exhaustivo de tales funciones, exhibamos su manejo con Maple:
En primer lugar, Maple conoce su definición en términos de la función exponencial. En primer lugar, el seno hiperbólico se define mediante
> convert(sinh(x),exp);
1
2
e
x
−
1
2
e
(
−x
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGLbWaaWbaaSqabeaacaWG4baaaOGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGLbWaaWbaaSqabeaadaqadaqaaiabgkHiTiaadIhaaiaawIcacaGLPaaaaaaaaa@4090@
Asimismo, el coseno hiperbólico se define mediante
> convert(cosh(x),exp);
1
2
e
x
+
1
2
e
(
−x
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGLbWaaWbaaSqabeaacaWG4baaaOGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacaWGLbWaaWbaaSqabeaadaqadaqaaiabgkHiTiaadIhaaiaawIcacaGLPaaaaaaaaa@4090@
Estas funciones recorren la hipérbola
>simplify(cosh(x)^2-sinh(x)^2,trig);
1
Maple conoce que
tanh(x)=
sinh(x)
cosh(x)
csch(x)=
1
sinh(x)
sech(x)=
1
cosh(x)
coth(x)=
1
tanh(x)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8301@
De hecho,
> simplify(tanh(x)-sinh(x)/cosh(x),trig);
0
> simplify(sinh(x)*csch(x),trig);
1
> simplify(cosh(x)*sech(x),trig);
1
> simplify(coth(x)*tanh(x),trig);
1
En consecuencia,
> convert(tanh(x),exp);
e
x
−
e
(
−x
)
e
x
+
e
(
−x
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGLbWaaWbaaSqabeaacaWG4baaaOGaeyOeI0IaamyzamaaCaaaleqabaWaaeWaaeaacqGHsislcaWG4baacaGLOaGaayzkaaaaaaGcbaGaamyzamaaCaaaleqabaGaamiEaaaakiabgUcaRiaadwgadaahaaWcbeqaamaabmaabaGaeyOeI0IaamiEaaGaayjkaiaawMcaaaaaaaaaaa@4526@
> convert(coth(x),exp);
e
x
+
e
(
−x
)
e
x
−
e
(
−x
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGLbWaaWbaaSqabeaacaWG4baaaOGaey4kaSIaamyzamaaCaaaleqabaWaaeWaaeaacqGHsislcaWG4baacaGLOaGaayzkaaaaaaGcbaGaamyzamaaCaaaleqabaGaamiEaaaakiabgkHiTiaadwgadaahaaWcbeqaamaabmaabaGaeyOeI0IaamiEaaGaayjkaiaawMcaaaaaaaaaaa@4526@
> convert(csch(x),exp);
2
e
x
−
e
(
−x
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaaabaGaamyzamaaCaaaleqabaGaamiEaaaakiabgkHiTiaadwgadaahaaWcbeqaamaabmaabaGaeyOeI0IaamiEaaGaayjkaiaawMcaaaaaaaaaaa@3E4E@
> convert(sech(x),exp);
2
e
x
+
e
(
−x
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaaabaGaamyzamaaCaaaleqabaGaamiEaaaakiabgkHiTiaadwgadaahaaWcbeqaamaabmaabaGaeyOeI0IaamiEaaGaayjkaiaawMcaaaaaaaaaaa@3E4E@
Por tanto, derivación e integración simbólica y numérica no son ya un secreto para Maple
> diff(sinh(x),x);
cosh(x)
> int(sinh(x),x);
cosh(x)
> int(cosh(x),x=0..1.);
Nótese el "." que sigue al 1, indicando cálculo en coma flotante
1.175201194
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