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hyperbolic: THEORY
%------------------------------------------------------------------------
% Definition of Hyperbolic Trig Functions
%
% Version 1.0 12/3/03
% Version 1.1 8/25/04
% Version 1.2 10/27/04 added exp_approx, ln_approx
%
% Author: David Lester
%
% Formula labels are from Handbook of Mathematical Functions
% by Abramowitz and Stegun
%------------------------------------------------------------------------
BEGIN
IMPORTING reals@sq, reals@sqrt,
ln_exp, %% RWB
reals@binomial
IMPORTING taylor_help, reals@polynomials
posreal_ge1: NONEMPTY_TYPE = {x:real | x >= 1 } CONTAINING 1
posreal_gt1: NONEMPTY_TYPE = {x:real | x > 1 } CONTAINING 2
posreal_le1: NONEMPTY_TYPE = {x:posreal | x <= 1 } CONTAINING 1/2
real_abs_lt1: NONEMPTY_TYPE = {x:real | -1 < x & x < 1} CONTAINING 0
real_abs_gt1: NONEMPTY_TYPE = {x:real | x < -1 OR 1 < x} CONTAINING 2
x,y: VAR real
pxle1: VAR posreal_le1
pxge1: VAR posreal_ge1
xalt1: VAR real_abs_lt1
n0x,n0y: VAR nzreal
n,m: VAR nat
% A&S Section 4.5 Hyperbolic Functions
sinh(x: real) :real = (exp(x)-exp(-x))/2 % 4.5.1
cosh(x: real) :posreal_ge1 = (exp(x)+exp(-x))/2 % 4.5.2
tanh(x: real) :real_abs_lt1 = sinh(x)/cosh(x) % 4.5.3
csch(n0x:nzreal):real = 1/sinh(n0x) % 4.5.4
sech(x :real) :posreal_le1 = 1/cosh(x) % 4.5.5
coth(n0x:nzreal):real_abs_gt1 = 1/tanh(n0x) % 4.5.6
% Restrictions for Branch Properties
nnreal_cosh (nnx:nnreal):posreal_ge1 = cosh(nnx)
posreal_csch(px:posreal):posreal = csch(px)
nnreal_sech (nnx:nnreal):posreal_le1 = sech(nnx)
posreal_coth(px:posreal):posreal_gt1 = coth(px)
% Monotonicity Properties
sinh_strict_increasing: LEMMA strict_increasing?(sinh)
cosh_strict_increasing: LEMMA strict_increasing?(nnreal_cosh)
tanh_strict_increasing: LEMMA strict_increasing?(tanh)
csch_strict_decreasing: LEMMA strict_decreasing?(posreal_csch)
sech_strict_decreasing: LEMMA strict_decreasing?(nnreal_sech)
coth_strict_decreasing: LEMMA strict_decreasing?(posreal_coth)
% Special Values of the Hyperbolic Functions
sinh_0: LEMMA sinh(0) = 0 % 4.5.61
cosh_0: LEMMA cosh(0) = 1 % 4.5.61
tanh_0: LEMMA tanh(0) = 0 % 4.5.61
sech_0: LEMMA sech(0) = 1 % 4.5.61
% Relations between Hyperbolic Functions
cosh_sinh_one: LEMMA sq(cosh(x)) - sq(sinh(x)) = 1 % 4.5.16
tanh_sech_one: LEMMA sq(tanh(x)) + sq(sech(x)) = 1 % 4.5.17
coth_csch_one: LEMMA sq(coth(n0x)) - sq(csch(n0x)) = 1 % 4.5.18
cosh_plus_sinh: LEMMA cosh(x) + sinh(x) = exp(x) % 4.5.19
cosh_minus_sinh: LEMMA cosh(x) - sinh(x) = exp(-x) % 4.5.20
% Negative Angle FormulasMA FORALL (k:{i:nat| i<n}): C(n,k+1) = C(n,k)*((n-k)/(k+1))
sinh_neg: LEMMA sinh(-x) = -sinh(x) % 4.5.21
cosh_neg: LEMMA cosh(-x) = cosh(x) % 4.5.22
tanh_neg: LEMMA tanh(-x) = -tanh(x) % 4.5.23
csch_neg: LEMMA csch(-n0x) = -csch(n0x)
sech_neg: LEMMA sech(-x) = sech(x)
coth_neg: LEMMA coth(-n0x) = -coth(n0x)
% Addition Formulas
sinh_sum: LEMMA sinh(x+y) = sinh(x)*cosh(y) + cosh(x)*sinh(y) % 4.5.24
sinh_diff: LEMMA sinh(x-y) = sinh(x)*cosh(y) - cosh(x)*sinh(y)
cosh_sum: LEMMA cosh(x+y) = cosh(x)*cosh(y) + sinh(x)*sinh(y) % 4.5.25
cosh_diff: LEMMA cosh(x-y) = cosh(x)*cosh(y) - sinh(x)*sinh(y)
tanh_sum: LEMMA tanh(x+y) = (tanh(x)+tanh(y))/(1+tanh(x)*tanh(y)) % 4.5.26
coth_sum: LEMMA n0x+n0y /= 0 IMPLIES % 4.5.27
coth(n0x+n0y)
= (1+coth(n0x)*coth(n0y))/(coth(n0x)+coth(n0y))
% Half-angle Formulas
sinh_half: LEMMA sinh(x/2) = LET y = sqrt((cosh(x)-1)/2) IN % 4.5.28
IF x >= 0 THEN y ELSE -y ENDIF
cosh_half: LEMMA cosh(x/2) = sqrt((cosh(x)+1)/2) % 4.5.29
tanh_half1: LEMMA tanh(x/2) = LET y = sqrt((cosh(x)-1)/(cosh(x)+1)) % 4.5.30
IN IF x >= 0 THEN y ELSE -y ENDIF
tanh_half2: LEMMA tanh(n0x/2) = (cosh(n0x)-1)/sinh(n0x) % 4.5.30
tanh_half3: LEMMA tanh(x/2) = sinh(x)/(cosh(x)+1) % 4.5.30
% Multiple-angle Formulas
sinh2x: LEMMA sinh(2*x) = 2*sinh(x)*cosh(x) % 4.5.31
sinh2x_B: LEMMA sinh(2*x) = 2*tanh(x)/(1-sq(tanh(x))) % 4.5.31
cosh2x: LEMMA cosh(2*x) = 2*sq(cosh(x))-1 % 4.5.32
cosh2x_B: LEMMA cosh(2*x) = 2*sq(sinh(x))+1 % 4.5.32
cosh2x_C: LEMMA cosh(2*x) = sq(cosh(x)) + sq(sinh(x)) % 4.5.32
tanh2x: LEMMA tanh(2*x) = 2*tanh(x)/(1+sq(tanh(x))) % 4.5.33
sinh3x: LEMMA sinh(3*x) = 3*sinh(x) + 4*sinh(x)^3 % 4.5.34
cosh3x: LEMMA cosh(3*x) = 4*cosh(x)^3-3*cosh(x) % 4.5.35
sinh4x: LEMMA sinh(4*x) % 4.5.36
= 4*sinh(x)*cosh(x)*(sq(sinh(x)) + sq(cosh(x)))
cosh4x: LEMMA cosh(4*x) % 4.5.37
= cosh(x)^4+6*sq(sinh(x)*cosh(x))+sinh(x)^4
% Products
sinh_times_sinh: LEMMA sinh(x)*sinh(y) = (cosh(x+y)-cosh(x-y))/2 % 4.5.38
cosh_times_cosh: LEMMA cosh(x)*cosh(y) = (cosh(x+y)+cosh(x-y))/2 % 4.5.39
sinh_times_cosh: LEMMA sinh(x)*cosh(y) = (sinh(x+y)+sinh(x-y))/2 % 4.5.40
% Addition and Subtraction
sum_sinh: LEMMA sinh(x)+sinh(y) = 2*sinh((x+y)/2)*cosh((x-y)/2) % 4.5.41
diff_sinh: LEMMA sinh(x)-sinh(y) = 2*cosh((x+y)/2)*sinh((x-y)/2) % 4.5.42
sum_cosh: LEMMA cosh(x)+cosh(y) = 2*cosh((x+y)/2)*cosh((x-y)/2) % 4.5.43
diff_cosh: LEMMA cosh(x)-cosh(y) = 2*sinh((x+y)/2)*sinh((x-y)/2) % 4.5.44
sum_tanh: LEMMA tanh(x)+tanh(y) = sinh(x+y)/(cosh(x)*cosh(y)) % 4.5.45
sum_coth: LEMMA coth(n0x)+coth(n0y) % 4.5.46
= sinh(n0x+n0y)/(sinh(n0x)*sinh(n0y))
% Relations between squares of hyperbolic sines and cosines
diff_sinh_sq: LEMMA sq(sinh(x))-sq(sinh(y)) = sinh(x+y)*sinh(x-y) % 4.5.47
diff_cosh_sq: LEMMA sq(cosh(x))-sq(cosh(y)) = sinh(x+y)*sinh(x-y) % 4.5.47
sum_cosh_sinh_sq: LEMMA sq(sinh(x))+sq(cosh(y)) % 4.5.48
= cosh(x+y)*cosh(x-y)
% De Moivre's Theorem
hyperbolic_deMoivre: LEMMA (cosh(x)+sinh(x))^n = cosh(n*x)+sinh(n*x) % 4.5.53
% Series expansions
sinh_series_n(x:real,n:nat):real
= sigma(0,n,LAMBDA (i:nat): x^(2*i+1)/factorial(2*i+1))
sinh_taylors: AXIOM EXISTS (c: between[real](0,x)): % 4.5.62
sinh(x) = sinh_series_n(x,n) + cosh(c)*x^(2*n+3)/factorial(2*n+3)
% Logarithmic representations of inverse hyperbolics
asinh(x:real): real = ln(x+sqrt(sq(x)+1)) % 4.6.20
acosh(x:posreal_ge1):nnreal = ln(x+sqrt(sq(x)-1)) % 4.6.21
atanh(x:real_abs_lt1):real = ln((1+x)/(1-x))/2 % 4.6.22
% Bijection relations
sinh_bij: LEMMA bijective?[real,real](sinh)
cosh_bij: LEMMA bijective?[nnreal,posreal_ge1](nnreal_cosh)
tanh_bij: LEMMA bijective?[real,real_abs_lt1](tanh)
csch_bij: LEMMA bijective?[posreal,posreal](posreal_csch)
sech_bij: LEMMA bijective?[nnreal,posreal_le1](nnreal_sech)
coth_bij: LEMMA bijective?[posreal,posreal_gt1](posreal_coth)
asinh_alt_def: LEMMA asinh(x) = inverse(sinh)(x)
asinh_sinh: LEMMA asinh(sinh(x)) = x
sinh_asinh: LEMMA sinh(asinh(x)) = x
asinh_strict_increasing: LEMMA strict_increasing?(asinh)
asinh_bij: LEMMA bijective?[real,real](asinh)
% Taylor Series for atanh
z: VAR real_abs_lt1
pn: VAR posnat
atanhF(n:nat)(i:nat):int
= IF i > 2*n OR odd?(i) THEN 0 ELSE factorial(2*n)*C(2*n+1,i) ENDIF
atanhD(n:nat)(x:real):real = (1-sq(x))^(2*n+1)
atanhN(n:nat):[real->real] = polynomial(atanhF(n),2*n)
% atanh_taylors_prep1: LEMMA
% derivable_n_times(atanhN(n),m) AND derivable_n_times(atanhD(n),m)
% atanh_taylors_prep2: LEMMA
% deriv(atanhN(n))
% = IF n = 0 THEN const_fun(0)
% ELSE polynomial(LAMBDA (i:nat): (i+1)*atanhF(n)(i+1),2*n-1)
% ENDIF
% atanh_taylors_prep3: LEMMA
% deriv(atanhD(n)) = (LAMBDA (z:real): -(4*n+2)*z*
% IF n = 0 THEN 1 ELSE (1-sq(z))*atanhD(n-1)(z) ENDIF)
% atanh_taylors_prep4: LEMMA
% deriv(deriv(atanhN(n)))
% = IF n = 0 THEN const_fun(0)
% ELSE polynomial(LAMBDA (i:nat): (i+2)*(i+1)*atanhF(n)(i+2),2*n-2)
% ENDIF
% atanh_taylors_prep5: LEMMA FORALL (f:[real_abs_lt1->real],
% g:[real_abs_lt1->nzreal]):
% derivable[real_abs_lt1](f) AND derivable[real_abs_lt1](g) IMPLIES
% derivable[real_abs_lt1](f/g^pn) AND
% deriv[real_abs_lt1](f/g^pn)
% = (deriv[real_abs_lt1](f)*g-pn*f*deriv[real_abs_lt1](g))/(g^(pn+1))
atanhND(n:nat):[real_abs_lt1->real]
= (LAMBDA (x:real_abs_lt1): atanhN(n)(x)/atanhD(n)(x))
% atanh_taylors_prep6: LEMMA %% Proof streamlined by Ben Di Vito %%
% nderiv[real_abs_lt1](2,LAMBDA (x:real_abs_lt1): atanhN(n)(x)/atanhD(n)(x))
% = (LAMBDA (x:real_abs_lt1): atanhN(n+1)(x)/atanhD(n+1)(x))
% atanh_taylors_prep7: LEMMA
% derivable_n_times[real_abs_lt1](atanhND(n),2*m)
% atanh_taylors_prep8: LEMMA nderiv[real_abs_lt1](2*m,atanhND(n))=atanhND(n+m)
% atanh_nderiv: LEMMA
% nderiv[real_abs_lt1](n,atanh)
% = IF n = 0 THEN atanh
% ELSIF even?(n) THEN deriv[real_abs_lt1](atanhND(n/2-1))
% ELSE atanhND((n-1)/2) ENDIF
% atanh_nderiv_0: LEMMA
% nderiv[real_abs_lt1](n,atanh)(0)
% = IF even?(n) THEN 0 ELSE factorial(n-1) ENDIF
% atanh_n_times_derivable: LEMMA derivable_n_times[real_abs_lt1](atanh,n)
atanh_series_term(z:real_abs_lt1):[nat->real]
= (LAMBDA (n:nat): z^(2*n+1)/(2*n+1))
atanh_series_n(z:real_abs_lt1,n:nat):real = sigma(0,n,atanh_series_term(z))
% atanh_series_eqv: LEMMA
% atanh_series_n(z,n) = sigma(0,2*n+2,
% LAMBDA (nn:nat): IF nn > 2*n+2 OR nn = 0 THEN 0 ELSE
% nderiv[real_abs_lt1](nn,atanh)(0)*
% z^nn/factorial(nn) ENDIF)
% atanh_taylors: LEMMA (EXISTS (c: between[real_abs_lt1](0,z)):
% atanh(z) = atanh_series_n(z,n) +
% nderiv[real_abs_lt1](2*n+3,atanh)(c)*
% z^(2*n+3)/factorial(2*n+3))
atanh_series: AXIOM abs(atanh(z)-atanh_series_n(z,n)) <=
((1+z)^(2*n+3)+(1-z)^(2*n+3))/(2*(2*n+3)*(1-sq(z))^(2*n+3))
END hyperbolic
% analysis@sqrt_derivative, analysis@restrict2_deriv,
% series@nth_derivatives, series@taylors,
% trig_fnd@atan,
% trig_fnd@polynomials,
% ln_exp_series,
% Derivatives
% sinh_derivable2: LEMMA derivable(sinh)
% cosh_derivable2: LEMMA derivable(cosh)
% tanh_derivable2: LEMMA derivable(tanh)
% deriv_sinh: LEMMA deriv(sinh) = cosh % 4.5.71
% deriv_cosh: LEMMA deriv(cosh) = sinh % 4.5.72
% deriv_tanh: LEMMA deriv(tanh) = sech*sech % 4.5.73
% Derivatives
% asinh_derivable2: LEMMA derivable(asinh)
% acosh_derivable2: LEMMA
% derivable[posreal_gt1](LAMBDA (x:posreal_gt1): acosh(x))
% atanh_derivable2: LEMMA derivable[real_abs_lt1](atanh)
%
% deriv_asinh: LEMMA deriv(asinh) % 4.6.37
% = (LAMBDA (x:real): 1/sqrt(1+sq(x)))
% deriv_acosh: LEMMA % 4.6.38%
% deriv[posreal_gt1](LAMBDA (x:posreal_gt1): acosh(x))
% = (LAMBDA (x:posreal_gt1): 1/sqrt(sq(x)-1))
% deriv_atanh: LEMMA deriv[real_abs_lt1](atanh) % 4.6.39
% = (LAMBDA (x:real_abs_lt1): 1/(1-sq(x)))
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