index

Wiki Index

Table of contents for the Introductio in analysin infinitorum wiki.

Chapter summaries

• chapter-1-on-functions-in-general — Euler’s definitional chapter: variables, functions, and their classification.
• chapter-2-on-the-transformation-of-functions — factoring polynomials and partial-fraction decomposition.
• chapter-3-on-the-transformation-of-functions-by-substitution — removing radicals, and rational parametrizations via $y=xz$.
• chapter-4-on-the-development-of-functions-in-infinite-series — expansion of rational and irrational functions as infinite series.
• chapter-5-on-functions-of-two-or-more-variables — several independent variables; homogeneous functions and the $y=uz$ reduction.
• chapter-6-on-exponentials-and-logarithms — first transcendental functions: $a^z$, $\log y$, transcendence of logs, geometric-mean tables, characteristic and mantissa.
• chapter-7-on-exponentials-and-logarithms-expressed-through-series — power series for $a^z$ and $\log (1+x)$ via $\omega$ infinitesimal and $j$ infinite; definition of $e$.
• chapter-8-on-transcendental-quantities-which-arise-from-the-circle — sine/cosine on the unit circle, De Moivre, the trig power series, $e^{iv}=\cos v+i\sin v$, the arctangent series, fast computation of $\pi$.
• chapter-9-on-trinomial-factors — constructive real factorization via $p^2-2pqz\cos \varphi +q^2{z}^2$; cyclotomic factorization of $a^n\pm z^n$; infinite products for $\sin z$, $\cos z$, $e^x\pm e^{-x}$.
• chapter-10-on-the-use-of-the-discovered-factors-to-sum-infinite-series — Newton’s identities applied to the Chapter 9 products: Basel problem, even-zeta values, character-style series, partial-fraction expansions of cot, csc, coth, csch.
• chapter-11-on-other-infinite-expressions-for-arcs-and-sines — linear-factor sine/cosine products at rational angles; Wallis product; log $\pi$ and log sin/cos by transposing double sums whose columns are even-zeta values.
• chapter-12-on-the-development-of-real-rational-functions — extension of partial-fraction decomposition to real quadratic (trinomial-factor) denominators; iterative algorithm for repeated quadratic factors.
• chapter-13-on-recurrent-series — closed-form general term and sum of any recurrent series from real partial fractions; De Moivre’s scale of the relation and its inversion.
• chapter-14-on-the-multiplication-and-division-of-angles — multiple-angle polynomials for sin/cos; factored products $\sin nz=2^{n-1}\sin z\cdots$; partial-fraction sums for csc, sec, cot, tan at equally-spaced angles; tangent of multiple angles via De Moivre; sums of trig functions in arithmetic progression (infinite and finite); inversion to powers of sin/cos as multiple-angle sums.
• chapter-15-on-series-which-arise-from-products — products $\prod (1\pm {\alpha }_{i}z{)}^{\pm 1}$ over primes give squarefree, Möbius, and Euler-product series; $\sum 1/p=\infty$ via logs at $n=1$; numerical table of $\sum 1/p^n$; prime-classified series for $\pi /4$, $\pi /2$, $\pi /(2\sqrt{2})$, $\pi /(3\sqrt{3})$ via the sieve of Eratosthenes lifted to series level.
• chapter-16-on-the-partition-of-numbers — partitions of integers via the products $\prod (1+x^{{\alpha }_i}z)$ (distinct parts) and $1/\prod (1-x^{{\alpha }_i}z)$ (with repetition); recurrent series for the column generating functions with triangular numerators; the Pascal-like recurrence $p_m(n)=p_{m-1}(n)+p_m(n-m)$ for the partition table; Euler’s pentagonal number theorem $\prod (1-x^k)=\sum (-1{)}^n{x}^{n(3n-1)/2}$; distinct = odd identity; uniqueness of binary and balanced-ternary representations.
• chapter-17-using-recurrent-series-to-find-roots-of-equations — Daniel Bernoulli’s method: from $x^m-\alpha x^{m-1}-\beta x^{m-2}-\cdots =0$, run a recurrent series with scale $\alpha ,\beta ,\gamma ,\dots$; the ratio $Q/P$ converges to the largest root in absolute value. Substitution shift $x=y+k$ makes the method universal; failure modes (close roots, $\pm p$ pairs, repeated roots); §348–§352 closed-form recovery of modulus and argument of a dominant complex pair from four consecutive coefficients; §354 geometric-progression interpretation; §355 partial application to transcendental equations.
• chapter-18-on-continued-fractions — the third kind of infinite expression; convergent three-term recurrence and alternation; bidirectional dictionary with alternating series giving Brouncker’s $4/\pi$, a CF for $\log 2$, and the celebrated $(e-1)/2=[0;1,6,10,14,18,\dots ]$ with arithmetic-progression quotients; periodic CFs = quadratic irrationals; rational CFs = Euclidean algorithm; best-rational-approximation principle applied to $\pi \approx 355/113$ and to the Gregorian leap-year rule.

Concepts

From Chapter 1

• function — Euler’s “analytic expression” definition.
• variable-and-constant — notation and the scope of a variable (including complex numbers).
• classification-of-functions — algebraic vs. transcendental, polynomial/rational/irrational.
• single-valued-and-multi-valued-functions — n-valued functions and Vieta-style relations.
• vietas-formulas — coefficient ↔ elementary-symmetric-function dictionary; the workhorse identification used in Chapters 1, 9, 10, 14.
• even-and-odd-functions — parity and its multiplicative rules.
• similar-functions — Euler’s template notion of “the same function of different variables.”

From Chapter 2

• factoring-polynomials — linear/quadratic factors, roots, leading-coefficient form.
• fundamental-theorem-of-algebra — Euler’s §32 claim that every real polynomial factors into real linear and quadratic factors.
• complex-conjugate-factors — pairing complex linear factors into real quadratics.
• intermediate-value-property — Euler’s informal IVT for polynomials.
• real-roots-by-degree-parity — existence of real roots from degree parity and constant-term sign.
• improper-rational-function — polynomial-part-plus-proper-remainder split.
• partial-fraction-decomposition — the §39–§46 algorithm, with the cover-up shortcut and repeated-factor tower.

From Chapter 3

• substitution — the technique: introduce a new variable $x$ defining both $y$ and $z$.
• rationalizing-substitutions — §47–§51 catalog of substitutions that remove radicals.
• rational-parametrization-of-the-circle — the §46/§50 half-angle parametrization.
• homogeneous-substitution — the §52–§58 $y=xz$ (or $y=x^m{z}^n$) trick.
• folium-of-descartes — Euler’s §52 worked example: $y^3+z^3=cyz$ parametrized rationally.

From Chapter 4

• geometric-series — §60: the series for $a/(\alpha +\beta z)$.
• method-of-undetermined-coefficients — §60–§61: positing a series and matching powers of $z$.
• recurrent-series — §62–§70: De Moivre’s recurrent series, law read off the denominator.
• higher-order-arithmetic-progressions — §64–§67: progressions with constant $k$-th differences as $(1-z{)}^{k+1}$-recurrent series.
• binomial-series — §71–§76: Newton’s $(P+Q{)}^{m/n}$ theorem and its recurrent form.

From Chapter 5

• functions-of-several-variables — §77–§82: independent variables, the old classification carried over, the equation-counting rule.
• implicit-function-and-equation-counting — §82: relations define variables implicitly; each independent equation removes one degree of freedom.
• homogeneous-function — §83–§91: degree of multivariate functions; Euler’s $y=uz$ reduction $V=z^{n}f(u)$; linear factorization of bivariate homogeneous polynomials.
• heterogeneous-function — §92–§93: bifid, trifid, and reduction to homogeneity by substitution.
• order-of-a-polynomial — §94: the greatest degree of any term, used to classify algebraic curves.
• reducible-polynomial — §94–§95: order of a polynomial and reducibility into non-irrational factors.

From Chapter 6

• exponential-function — §96–§101: $a^z$ defined by extension from integers to reals; canonical case $a>1$.
• logarithm — §102–§104: inverse of $a^z$; the four algebraic rules.
• transcendence-of-logarithms — §105: logs of “generic” rationals are transcendental.
• geometric-mean-method-for-logarithms — §106: iterated $\sqrt{AB}$ algorithm used by Briggs and Vlacq.
• change-of-base — §107–§108: a single multiplicative constant converts any log table to any other base.
• common-logarithm — §112: base-10 logs and their special role in decimal computation.
• characteristic-and-mantissa — §112–§113: integer/fractional split of base-10 logs; digit count from characteristic, digit string from mantissa.

From Chapter 7

• infinitesimal-and-infinite-numbers — Euler’s working device: $\omega$ infinitely small, $j$ infinitely large, $j\omega =z$ finite; the collapse $(j-m)/(nj)=1/n$.
• exponential-series — §115–§117: $a^z=\sum (kz{)}^n/n!$ from $(1+kz/j{)}^j$; the general $b^z$ via $\log b$.
• logarithmic-series — §118–§121: $\log (1+x)=(1/k)(x-x^2/2+\cdots )$, the §120 divergence paradox at $a=10$, and the fast-converging $\log \frac{1+x}{1-x}$ variant.
• eulers-number — §122: the base for which $k=1$, giving $e=\sum 1/n!=2.71828\dots$ — first appearance of the symbol $e$.
• natural-logarithm — §123–§125: ${\log }_e$, the integer table to twenty digits, and $k={\log }_{e}a$ as the universal change-of-base factor.

From Chapter 8

• pi — §126: the symbol $\pi$ enters notation; 113-digit decimal of half the unit circumference.
• sine-and-cosine — §127: definitions on the unit circle, special values, Pythagorean identity, co-function relations, tangent and cotangent.
• trigonometric-addition-formulas — §128, §130, §131: sum/difference, periodicity catalog, product-to-sum, sum-to-product, half-angle.
• trigonometric-recurrent-progression — §129: arcs in arithmetic progression have sines and cosines forming a recurrent series with denominator $1-2\cos y\cdot Z+Z^2$.
• de-moivre-formula — §132–§133: $(\cos z\pm i\sin z{)}^n=\cos nz\pm i\sin nz$; binomial expansions of $\cos nz$ and $\sin nz$.
• sine-and-cosine-series — §134: $\cos v=\sum (-1{)}^k{v}^{2k}/(2k)!$, $\sin v=\sum (-1{)}^k{v}^{2k+1}/(2k+1)!$ from De Moivre under $z$ infinitesimal, $n$ infinite, $nz=v$ finite.
• eulers-formula — §138: $e^{iv}=\cos v+i\sin v$; sines and cosines as complex exponentials.
• arctangent-series — §139–§141: $z=(1/2i)\log \frac{1+i\tan z}{1-i\tan z}$, $\arctan t=t-t^3/3+t^5/5-\cdots$, Leibniz $\pi /4$, $\pi$ from $\arctan (1/\sqrt{3})$.
• machin-like-formula — §142: $\pi /4=\arctan (1/2)+\arctan (1/3)$ for fast rational computation of $\pi$.

From Chapter 9

• trinomial-factor — §144–§146: real quadratic factor $p^2-2pqz\cos \varphi +q^2{z}^2$ from a complex linear conjugate pair.
• factorization-of-an-plus-minus-zn — §150–§153: cyclotomic factorization of $a^n+z^n$, $a^n-z^n$, and $a^{2n}-2a^n{z}^n\cos g+z^{2n}$.
• exponential-infinite-product — §155–§157: infinite products for $e^x-1$, $(e^x-e^{-x})/2$, $(e^x+e^{-x})/2$ from $(1+x/j{)}^j$.
• sine-infinite-product — §158: $\sin z=z\prod (1-z^2/k^2{\pi }^2)$ from $x=iz$ in the §156 product.
• cosine-infinite-product — §158: $\cos z=\prod (1-4z^2/(2k+1{)}^2{\pi }^2)$ from $x=iz$ in the §157 product.

From Chapter 10

• newtons-identities — §165–§166: the recurrence $P=A$, $Q=AP-2B$, $R=AQ-BP+3C$, $\dots$ converting elementary symmetric coefficients into power sums.
• basel-problem — §167: $\sum 1/k^2={\pi }^2/6$ via the sinh product and Newton’s identities.
• zeta-at-even-integers — §167–§169: tabulation of $\zeta (2k)$ as a rational multiple of ${\pi }^{2k}$ through $\zeta (26)$; sums over odd squares from the cosh product.
• odd-and-alternating-zeta-decomposition — §170: $M$, $M/2^n$, $M-M/2^n$, $M-2M/2^n$ — the elementary algebra giving even, odd, and alternating restrictions.
• circular-arc-series — §171–§180: applying Newton’s identities to the §164 arc-form products yields the Leibniz $\pi /4$ family and a vast catalog of character-style sums.
• cotangent-partial-fraction — §181–§183: partial-fraction expansions of $\pi \cot (\pi \sqrt{a})$, $\pi /\sin (\pi \sqrt{a})$ and their hyperbolic counterparts.

From Chapter 11

• linear-factors-of-sine-cosine — §184: the §158 sine/cosine products at $z=m\pi /(2n)$ split each quadratic factor as $(kn-m)/kn\cdot (kn+m)/kn$; co-function identity gives a second product per function.
• wallis-product — §185: $\pi /2=(2\cdot 2\cdot 4\cdot 4\cdot 6\cdot 6\cdots )/(1\cdot 3\cdot 3\cdot 5\cdot 5\cdot 7\cdots )$ from dividing the two §184 expressions, plus parametric variants for $\sqrt{2}$ and others.
• trig-infinite-products — §186–§187: linear-factor infinite products for $\tan ,\cot ,\sec ,\csc$ as quotients; ratio formulas for $\sin (m\pi /2n)/\sin (k\pi /2n)$.
• log-pi-via-products — §188–§190: $\log \pi =\log 4-(A-1)-(B-1)/2-\cdots$ via taking logs of the Wallis product, expanding by §118, and transposing.
• log-sine-via-products — §191–§198: $\log \sin (m\pi /2n)$ and $\log \cos (m\pi /2n)$ by the same transposition, sharing the table $A,B,C,\dots$ and $\alpha ,\beta ,\gamma ,\dots$ with $\log \pi$; §197 fast tan/cot via the §181 partial fraction.

From Chapter 12

• real-partial-fraction-decomposition — §199–§210: real partial fractions for a rational function with real quadratic factors $p^2-2pqz\cos \varphi +q^2{z}^2$ in the denominator; closed-form coefficients via De Moivre substitution at the complex roots; iterative tower for repeated quadratic factors.

From Chapter 13

• general-term-of-recurrent-series — §211–§223: closed-form coefficient of $z^n$ via real partial fractions; linear-factor brick $A/(1-pz{)}^k$ gives binomial-times-power, quadratic-factor brick gives $\sin (n+1)\varphi /\sin \varphi$.
• scale-of-the-relation — §224: De Moivre’s name for the recurrence multipliers $\alpha ,\beta ,\gamma ,\dots$; equivalent to the (sign-flipped) denominator of the generating rational function.
• closed-form-two-term-recurrence — §226–§229: Binet-type $X_n=U{p}^n+V{q}^n$ for two-member scales; the invariant $UV=(B^2-\alpha AB+\beta A^2)/(4\beta -{\alpha }^2)$; term from a single predecessor by an “illusory” square root.
• sum-of-recurrent-series — §231–§233: infinite sum equals the generating rational function; partial sum collapses to first/last terms only for two-member scales.

From Chapter 14

• multiple-angle-polynomials — §234–§238, §243: recurrence-generated tables for $\sin nz$ and $\cos nz$; pure polynomial in $\sin z$ for odd $n$; residual $\sqrt{1-x^2}$ factor for even $n$; Chebyshev polynomials in disguise.
• trig-values-as-roots — §235–§256: the $n$ roots of the multiple-angle polynomial are trig functions at equally-spaced angles; Vieta’s formulas generate all partial-fraction and product identities.
• sine-cosine-factored-products — §237, §240–§245: $\sin nz=2^{n-1}\sin z\cdot \sin (\pi /n-z)\sin (\pi /n+z)\cdots$; unified formula for odd and even $n$; matching cosine products.
• trig-multiple-angle-partial-fractions — §237, §246–§256: $n/\sin nz$, $n\cot nz$, $n\sec nz$, $n\csc nz$ each as a sum of $n$ same-function evaluations at shifted angles; tangent/cotangent products from De Moivre.
• sum-of-trig-in-ap — §258–§260: closed-form for the infinite sum $\sin a+\sin (a+b)+\cdots =(\sin a-\sin (a-b))/(2-2\cos b)$ via the recurrent-series generating function at $z=1$, plus the standard finite-AP formula by subtracting the tail.
• powers-of-sine-and-cosine — §261–§263: inversion of the multiple-angle expansion: $(\sin z{)}^n$ and $(\cos z{)}^n$ as binomial-weighted finite sums of $\sin kz$, $\cos kz$.

From Chapter 15

• euler-product-formula — §270–§277, §283–§284: $\sum 1/k^n={\prod }_p(1-1/p^n{)}^{-1}$, derived both by expanding the reciprocal product (unique factorisation) and by an Eratosthenes-style sieve on the series; reciprocal Möbius relation.
• squarefree-and-mobius-series — §267–§269: $\prod (1+1/p^n)={\sum }_{k\text{ squarefree}}1/k^n$; the negative-factor version is the Möbius-signed series $\sum \mu (k)/k^n=1/\zeta (n)$.
• divergence-of-prime-reciprocals — §278–§280: $\sum 1/p=\infty$ via logarithm of the Euler product at $n=1$; quantitative refinement of Euclid’s theorem.
• prime-zeta-values — §281–§282: numerical table of $\sum 1/p^n$ for even $n=2$ to $36$ to 12-digit precision, obtained by inverting the §278 identity.
• prime-sign-series-for-pi — §285–§296: catalogue of series and products for $\pi /4$, $\pi /2$, ${\pi }^3/32$, $\pi /(3\sqrt{3})$, $\pi /(2\sqrt{2})$ with signs based on prime residues mod 4, mod 6, mod 8 — the Dirichlet $L$-functions of small conductor in Euler-product form, a century before Dirichlet.

From Chapter 16

• partition-of-numbers — §297–§331: the central concept of integer partitions; distinct vs. unrestricted; notation $p(n),p_m(n),q(n),q_m(n)$; index of the chapter’s theorems.
• partition-generating-functions — §297–§315: bivariate products $\prod (1+x^{{\alpha }_i}z)$ and $1/\prod (1-x^{{\alpha }_i}z)$ enumerating partitions by part-count $z^m$ and size $x^n$; functional-equation derivation of the closed-form $P_m$ with triangular-number numerators.
• partitions-into-distinct-parts — §299–§315: $q_m(n)$ generating function, recurrent series $x^{m(m+1)/2}/{\prod }_{k=1}^m(1-x^k)$, the staircase bijection $q_m(n)=p_m(n-m(m-1)/2)$.
• partition-recurrence — §316–§318: the Pascal-like rule $p_m(n)=p_{m-1}(n)+p_m(n-m)$ by which Euler’s partition table is filled column by column.
• eulers-pentagonal-number-theorem — §323–§324: $\prod (1-x^k)=\sum (-1{)}^n{x}^{n(3n-1)/2}$; the induced $O(\sqrt{n})$-term recurrence for $p(n)$; Euler’s empirical discovery (no proof in the Introductio).
• distinct-parts-equals-odd-parts — §325–§327: $\prod (1+x^k)=\prod 1/(1-x^{2k-1})$ via the one-line identity $PQ=\prod (1-x^{2k})$; partitions of $9$ as worked example; computation of $q(n)$ from $p(n)$.
• binary-representation-theorem — §328–§329: $\prod (1+x^{2^k})=1/(1-x)$ by fixed-point/functional equation; uniqueness of binary; weighing with $1,2,4,8,\dots$-pound weights.
• balanced-ternary-representation — §330–§331: $\prod (x^{-3^k}+1+x^{3^k})={\sum }_{n\in \mathbb{Z}}{x}^n$; every integer is uniquely a signed sum of distinct powers of $3$ with digits $\{-1,0,+1\}$; two-pan balance weighing.

From Chapter 17

• bernoullis-method-for-roots — §332–§347, §354–§355: from the equation’s coefficients form a recurrent series with scale $\alpha ,\beta ,\gamma ,\dots$; ratio $Q/P$ of consecutive coefficients converges to the largest root in absolute value; substitution shift $x=y+k$ makes any root findable as the smallest of a transformed equation; failure modes (close roots, $\pm p$ pairs, repeated roots); §354 geometric-progression interpretation; §355 partial application to $\sin z=1/2$.
• trinomial-factor-from-recurrent-series — §348–§353: when the dominant pole is a complex conjugate pair, $Q/P$ oscillates; Euler eliminates the unknowns and the index to recover $p=\sqrt{(R^2-QS)/(Q^2-PR)}$ and $\cos \varphi =(QR-PS)/(2\sqrt{(Q^2-PR)(R^2-QS)})$ from four consecutive coefficients, giving both modulus and argument of the dominant complex pair.

From Chapter 18

• continued-fraction — §357: the two forms (numerators all $1$, or arbitrary $\alpha ,\beta ,\gamma ,\dots$); the third kind of infinite expression after series and products.
• convergents-of-a-continued-fraction — §358–§362: three-term recurrence $N_k=b_k{N}_{k-1}+{\beta }_k{N}_{k-2}$ shared by numerators and denominators; the $1/0$ prefix; alternation of truncations around the true value.
• continued-fraction-series-equivalence — §363–§373: telescoping difference of convergents gives the alternating series $a+\alpha /P-\alpha \beta /(PQ)+\alpha \beta \gamma /(QR)-\cdots$; inverse templates converting alternating series back into CFs via free choice of partial denominators.
• brouncker-formula — §369 Example II: $4/\pi =1+1^2/(2+3^2/(2+5^2/(2+\cdots )))$, the first CF for $\pi$ in history, recovered as a special case of the §369 reciprocal-series template applied to the Leibniz series.
• continued-fraction-for-log-2 — §369 Example I: $\log 2=1/(1+1/(1+4/(1+9/(1+\cdots ))))$, partial numerators $1,1,4,9,16,25,\dots$ from the alternating-harmonic series.
• continued-fraction-for-e — §381 Example III: $(e-1)/2=[0;1,6,10,14,18,22,\dots ]$, partial quotients in arithmetic progression with common difference $4$ — Euler’s empirical discovery from the Euclidean algorithm on a 13-digit decimal.
• periodic-continued-fractions — §376–§379: periodic simple CFs satisfy quadratic equations, so represent quadratic irrationals; single-letter periods give $\sqrt{a^2+4}$; two-letter periods extend the method to all square roots; $\sqrt{7}\approx 2024/765$ with error $<3/10^7$.
• euclidean-algorithm-continued-fraction — §381: rational $A/B$‘s CF expansion = Euclidean-algorithm quotients of $A$ and $B$; applied to decimals it produces the CF of an irrational; recovers the $\sqrt{2}$ pattern of §376 and discovers the AP pattern for $e$.
• best-rational-approximations — §382: Wallis’s principle that the convergents are the best rational approximations with bounded denominator; $\pi \to 22/7,333/106,355/113$ (the famous Metian ratio); the solar-year computation yielding the Julian $1/4$ and Gregorian $97/400$ leap-day rules.