Theory Generalised_Binomial_Theorem
section ‹Generalised Binomial Theorem›
text ‹
The proof of the Generalised Binomial Theorem and related results.
We prove the generalised binomial theorem for complex numbers, following the proof at:
🌐‹https://proofwiki.org/wiki/Binomial_Theorem/General_Binomial_Theorem›
›
theory Generalised_Binomial_Theorem
imports
Complex_Main
Complex_Transcendental
Summation_Tests
begin
lemma gbinomial_ratio_limit:
fixes a :: "'a :: real_normed_field"
assumes "a ∉ ℕ"
shows "(λn. (a gchoose n) / (a gchoose Suc n)) ⇢ -1"
proof (rule Lim_transform_eventually)
let ?f = "λn. inverse (a / of_nat (Suc n) - of_nat n / of_nat (Suc n))"
from eventually_gt_at_top[of "0::nat"]
show "eventually (λn. ?f n = (a gchoose n) /(a gchoose Suc n)) sequentially"
proof eventually_elim
fix n :: nat assume n: "n > 0"
then obtain q where q: "n = Suc q" by (cases n) blast
let ?P = "∏i=0..<n. a - of_nat i"
from n have "(a gchoose n) / (a gchoose Suc n) = (of_nat (Suc n) :: 'a) *
(?P / (∏i=0..n. a - of_nat i))"
by (simp add: gbinomial_prod_rev atLeastLessThanSuc_atLeastAtMost)
also from q have "(∏i=0..n. a - of_nat i) = ?P * (a - of_nat n)"
by (simp add: prod.atLeast0_atMost_Suc atLeastLessThanSuc_atLeastAtMost)
also have "?P / … = (?P / ?P) / (a - of_nat n)" by (rule divide_divide_eq_left[symmetric])
also from assms have "?P / ?P = 1" by auto
also have "of_nat (Suc n) * (1 / (a - of_nat n)) =
inverse (inverse (of_nat (Suc n)) * (a - of_nat n))" by (simp add: field_simps)
also have "inverse (of_nat (Suc n)) * (a - of_nat n) = a / of_nat (Suc n) - of_nat n / of_nat (Suc n)"
by (simp add: field_simps del: of_nat_Suc)
finally show "?f n = (a gchoose n) / (a gchoose Suc n)" by simp
qed
have "(λn. norm a / (of_nat (Suc n))) ⇢ 0"
unfolding divide_inverse
by (intro tendsto_mult_right_zero LIMSEQ_inverse_real_of_nat)
hence "(λn. a / of_nat (Suc n)) ⇢ 0"
by (subst tendsto_norm_zero_iff[symmetric]) (simp add: norm_divide del: of_nat_Suc)
hence "?f ⇢ inverse (0 - 1)"
by (intro tendsto_inverse tendsto_diff LIMSEQ_n_over_Suc_n) simp_all
thus "?f ⇢ -1" by simp
qed
lemma conv_radius_gchoose:
fixes a :: "'a :: {real_normed_field,banach}"
shows "conv_radius (λn. a gchoose n) = (if a ∈ ℕ then ∞ else 1)"
proof (cases "a ∈ ℕ")
assume a: "a ∈ ℕ"
have "eventually (λn. (a gchoose n) = 0) sequentially"
using eventually_gt_at_top[of "nat ⌊norm a⌋"]
by eventually_elim (insert a, auto elim!: Nats_cases simp: binomial_gbinomial[symmetric])
from conv_radius_cong'[OF this] a show ?thesis by simp
next
assume a: "a ∉ ℕ"
from tendsto_norm[OF gbinomial_ratio_limit[OF this]]
have "conv_radius (λn. a gchoose n) = 1"
by (intro conv_radius_ratio_limit_nonzero[of _ 1]) (simp_all add: norm_divide)
with a show ?thesis by simp
qed
theorem gen_binomial_complex:
fixes z :: complex
assumes "norm z < 1"
shows "(λn. (a gchoose n) * z^n) sums (1 + z) powr a"
proof -
define K where "K = 1 - (1 - norm z) / 2"
from assms have K: "K > 0" "K < 1" "norm z < K"
unfolding K_def by (auto simp: field_simps intro!: add_pos_nonneg)
let ?f = "λn. a gchoose n" and ?f' = "diffs (λn. a gchoose n)"
have summable_strong: "summable (λn. ?f n * z ^ n)" if "norm z < 1" for z using that
by (intro summable_in_conv_radius) (simp_all add: conv_radius_gchoose)
with K have summable: "summable (λn. ?f n * z ^ n)" if "norm z < K" for z using that by auto
hence summable': "summable (λn. ?f' n * z ^ n)" if "norm z < K" for z using that
by (intro termdiff_converges[of _ K]) simp_all
define f f' where [abs_def]: "f z = (∑n. ?f n * z ^ n)" "f' z = (∑n. ?f' n * z ^ n)" for z
{
fix z :: complex assume z: "norm z < K"
from summable_mult2[OF summable'[OF z], of z]
have summable1: "summable (λn. ?f' n * z ^ Suc n)" by (simp add: mult_ac)
hence summable2: "summable (λn. of_nat n * ?f n * z^n)"
unfolding diffs_def by (subst (asm) summable_Suc_iff)
have "(1 + z) * f' z = (∑n. ?f' n * z^n) + (∑n. ?f' n * z^Suc n)"
unfolding f_f'_def using summable' z by (simp add: algebra_simps suminf_mult)
also have "(∑n. ?f' n * z^n) = (∑n. of_nat (Suc n) * ?f (Suc n) * z^n)"
by (intro suminf_cong) (simp add: diffs_def)
also have "(∑n. ?f' n * z^Suc n) = (∑n. of_nat n * ?f n * z ^ n)"
using summable1 suminf_split_initial_segment[OF summable1] unfolding diffs_def
by (subst suminf_split_head, subst (asm) summable_Suc_iff) simp_all
also have "(∑n. of_nat (Suc n) * ?f (Suc n) * z^n) + (∑n. of_nat n * ?f n * z^n) =
(∑n. a * ?f n * z^n)"
by (subst gbinomial_mult_1, subst suminf_add)
(insert summable'[OF z] summable2,
simp_all add: summable_powser_split_head algebra_simps diffs_def)
also have "… = a * f z" unfolding f_f'_def
by (subst suminf_mult[symmetric]) (simp_all add: summable[OF z] mult_ac)
finally have "a * f z = (1 + z) * f' z" by simp
} note deriv = this
have [derivative_intros]: "(f has_field_derivative f' z) (at z)" if "norm z < of_real K" for z
unfolding f_f'_def using K that
by (intro termdiffs_strong[of "?f" K z] summable_strong) simp_all
have "f 0 = (∑n. if n = 0 then 1 else 0)" unfolding f_f'_def by (intro suminf_cong) simp
also have "… = 1" using sums_single[of 0 "λ_. 1::complex"] unfolding sums_iff by simp
finally have [simp]: "f 0 = 1" .
have "∃c. ∀z∈ball 0 K. f z * (1 + z) powr (-a) = c"
proof (rule has_field_derivative_zero_constant)
fix z :: complex assume z': "z ∈ ball 0 K"
hence z: "norm z < K" by simp
with K have nz: "1 + z ≠ 0" by (auto dest!: minus_unique)
from z K have "norm z < 1" by simp
hence "(1 + z) ∉ ℝ⇩≤⇩0" by (cases z) (auto simp: Complex_eq complex_nonpos_Reals_iff)
hence "((λz. f z * (1 + z) powr (-a)) has_field_derivative
f' z * (1 + z) powr (-a) - a * f z * (1 + z) powr (-a-1)) (at z)" using z
by (auto intro!: derivative_eq_intros)
also from z have "a * f z = (1 + z) * f' z" by (rule deriv)
finally show "((λz. f z * (1 + z) powr (-a)) has_field_derivative 0) (at z within ball 0 K)"
using nz by (simp add: field_simps powr_diff at_within_open[OF z'])
qed simp_all
then obtain c where c: "⋀z. z ∈ ball 0 K ⟹ f z * (1 + z) powr (-a) = c" by blast
from c[of 0] and K have "c = 1" by simp
with c[of z] have "f z = (1 + z) powr a" using K
by (simp add: powr_minus field_simps dist_complex_def)
with summable K show ?thesis unfolding f_f'_def by (simp add: sums_iff)
qed
lemma gen_binomial_complex':
fixes x y :: real and a :: complex
assumes "¦x¦ < ¦y¦"
shows "(λn. (a gchoose n) * of_real x^n * of_real y powr (a - of_nat n)) sums
of_real (x + y) powr a" (is "?P x y")
proof -
{
fix x y :: real assume xy: "¦x¦ < ¦y¦" "y ≥ 0"
hence "y > 0" by simp
note xy = xy this
from xy have "(λn. (a gchoose n) * of_real (x / y) ^ n) sums (1 + of_real (x / y)) powr a"
by (intro gen_binomial_complex) (simp add: norm_divide)
hence "(λn. (a gchoose n) * of_real (x / y) ^ n * y powr a) sums
((1 + of_real (x / y)) powr a * y powr a)"
by (rule sums_mult2)
also have "(1 + complex_of_real (x / y)) = complex_of_real (1 + x/y)" by simp
also from xy have "… powr a * of_real y powr a = (… * y) powr a"
by (subst powr_times_real[symmetric]) (simp_all add: field_simps)
also from xy have "complex_of_real (1 + x / y) * complex_of_real y = of_real (x + y)"
by (simp add: field_simps)
finally have "?P x y" using xy by (simp add: field_simps powr_diff powr_nat)
} note A = this
show ?thesis
proof (cases "y < 0")
assume y: "y < 0"
with assms have xy: "x + y < 0" by simp
with assms have "¦-x¦ < ¦-y¦" "-y ≥ 0" by simp_all
note A[OF this]
also have "complex_of_real (-x + -y) = - complex_of_real (x + y)" by simp
also from xy assms have "... powr a = (-1) powr -a * of_real (x + y) powr a"
by (subst powr_neg_real_complex) (simp add: abs_real_def split: if_split_asm)
also {
fix n :: nat
from y have "(a gchoose n) * of_real (-x) ^ n * of_real (-y) powr (a - of_nat n) =
(a gchoose n) * (-of_real x / -of_real y) ^ n * (- of_real y) powr a"
by (subst power_divide) (simp add: powr_diff powr_nat)
also from y have "(- of_real y) powr a = (-1) powr -a * of_real y powr a"
by (subst powr_neg_real_complex) simp
also have "-complex_of_real x / -complex_of_real y = complex_of_real x / complex_of_real y"
by simp
also have "... ^ n = of_real x ^ n / of_real y ^ n" by (simp add: power_divide)
also have "(a gchoose n) * ... * ((-1) powr -a * of_real y powr a) =
(-1) powr -a * ((a gchoose n) * of_real x ^ n * of_real y powr (a - n))"
by (simp add: algebra_simps powr_diff powr_nat)
finally have "(a gchoose n) * of_real (- x) ^ n * of_real (- y) powr (a - of_nat n) =
(-1) powr -a * ((a gchoose n) * of_real x ^ n * of_real y powr (a - of_nat n))" .
}
note sums_cong[OF this]
finally show ?thesis by (simp add: sums_mult_iff)
qed (insert A[of x y] assms, simp_all add: not_less)
qed
lemma gen_binomial_complex'':
fixes x y :: real and a :: complex
assumes "¦y¦ < ¦x¦"
shows "(λn. (a gchoose n) * of_real x powr (a - of_nat n) * of_real y ^ n) sums
of_real (x + y) powr a"
using gen_binomial_complex'[OF assms] by (simp add: mult_ac add.commute)
lemma gen_binomial_real:
fixes z :: real
assumes "¦z¦ < 1"
shows "(λn. (a gchoose n) * z^n) sums (1 + z) powr a"
proof -
from assms have "norm (of_real z :: complex) < 1" by simp
from gen_binomial_complex[OF this]
have "(λn. (of_real a gchoose n :: complex) * of_real z ^ n) sums
(of_real (1 + z)) powr (of_real a)" by simp
also have "(of_real (1 + z) :: complex) powr (of_real a) = of_real ((1 + z) powr a)"
using assms by (subst powr_of_real) simp_all
also have "(of_real a gchoose n :: complex) = of_real (a gchoose n)" for n
by (simp add: gbinomial_prod_rev)
hence "(λn. (of_real a gchoose n :: complex) * of_real z ^ n) =
(λn. of_real ((a gchoose n) * z ^ n))" by (intro ext) simp
finally show ?thesis by (simp only: sums_of_real_iff)
qed
lemma gen_binomial_real':
fixes x y a :: real
assumes "¦x¦ < y"
shows "(λn. (a gchoose n) * x^n * y powr (a - of_nat n)) sums (x + y) powr a"
proof -
from assms have "y > 0" by simp
note xy = this assms
from assms have "¦x / y¦ < 1" by simp
hence "(λn. (a gchoose n) * (x / y) ^ n) sums (1 + x / y) powr a"
by (rule gen_binomial_real)
hence "(λn. (a gchoose n) * (x / y) ^ n * y powr a) sums ((1 + x / y) powr a * y powr a)"
by (rule sums_mult2)
with xy show ?thesis
by (simp add: field_simps powr_divide powr_diff powr_realpow)
qed
lemma one_plus_neg_powr_powser:
fixes z s :: complex
assumes "norm (z :: complex) < 1"
shows "(λn. (-1)^n * ((s + n - 1) gchoose n) * z^n) sums (1 + z) powr (-s)"
using gen_binomial_complex[OF assms, of "-s"] by (simp add: gbinomial_minus)
lemma gen_binomial_real'':
fixes x y a :: real
assumes "¦y¦ < x"
shows "(λn. (a gchoose n) * x powr (a - of_nat n) * y^n) sums (x + y) powr a"
using gen_binomial_real'[OF assms] by (simp add: mult_ac add.commute)
lemma sqrt_series':
"¦z¦ < a ⟹ (λn. ((1/2) gchoose n) * a powr (1/2 - real_of_nat n) * z ^ n) sums
sqrt (a + z :: real)"
using gen_binomial_real''[of z a "1/2"] by (simp add: powr_half_sqrt)
lemma sqrt_series:
"¦z¦ < 1 ⟹ (λn. ((1/2) gchoose n) * z ^ n) sums sqrt (1 + z)"
using gen_binomial_real[of z "1/2"] by (simp add: powr_half_sqrt)
end