樹莓派、樹莓派之學習、樹莓派之教育

時間序列︰生成函數‧漸近展開︰白努利 □○《四》

白努利數起源於等冪求和公式

Faulhaber’s formula

Theorem

Let n and p be positive integers.

Then:

\sum \limits_{k = 1}^n k^p = \frac 1 {p + 1} \sum \limits_{i = 0}^p \left({-1}\right)^i \binom {p + 1} i B_i n^{p + 1 - i}

where Bn denotes the nth Bernoulli number.

 

之符號通解的追求應無疑議。

從白努利《Ars Conjectandi》書中的形式表述看來

Reconstruction of “Summae Potestatum“

Jakob Bernoulli’s Summae Potestatum, 1713

The Bernoulli numbers were introduced by Jakob Bernoulli in the book Ars Conjectandi published posthumously in 1713 page 97. The main formula can be seen in the second half of the corresponding facsimile. The constant coefficients denoted A, B, C and D by Bernoulli are mapped to the notation which is now prevalent as A = B2, B = B4, C = B6, D = B8. The expression c·c−1·c−2·c−3 means c·(c−1)·(c−2)·(c−3) – the small dots are used as grouping symbols. Using today’s terminology these expressions are falling factorial powers ck. The factorial notation k! as a shortcut for 1 × 2 × … × k was not introduced until 100 years later. The integral symbol on the left hand side goes back to Gottfried Wilhelm Leibniz in 1675 who used it as a long letter S for “summa” (sum). (The Mathematics Genealogy Project[14] shows Leibniz as the doctoral adviser of Jakob Bernoulli. See also the Earliest Uses of Symbols of Calculus.[15]) The letter n on the left hand side is not an index of summation but gives the upper limit of the range of summation which is to be understood as 1, 2, …, n. Putting things together, for positive c, today a mathematician is likely to write Bernoulli’s formula as:

{\displaystyle \sum _{k=1}^{n}k^{c}={\frac {n^{c+1}}{c+1}}+{\frac {1}{2}}n^{c}+\sum _{k=2}^{\infty }{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}.}

In fact this formula imperatively suggests to set B1 = 1/2 when switching from the so-called ‘archaic’ enumeration which uses only the even indices 2, 4, 6… to the modern form (more on different conventions in the next paragraph). Most striking in this context is the fact that the falling factorial ck−1 has for k = 0 the value \frac{1}{c+1} .[16] Thus Bernoulli’s formula can and has to be written

{\displaystyle \sum _{k=1}^{n}k^{c}=\sum _{k=0}^{\infty }{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}}

if B1 stands for the value Bernoulli himself has given to the coefficient at that position.

 

恐尚未能將白努利數看成數列吧！何況泰勒級數

Taylor series

History

The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility: the result was Zeno’s paradox. Later, Aristotle proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by Archimedes, as it had been prior to Aristotle by the Presocratic Atomist Democritus. It was through Archimedes’s method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite result.[1] Liu Hui independently employed a similar method a few centuries later.[2]

In the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama.[3][4] Though no record of his work survives, writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series, including those for the trigonometric functions of sine, cosine, tangent, and arctangent. The Kerala school of astronomy and mathematics further expanded his works with various series expansions and rational approximations until the 16th century.

In the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor,[5] after whom the series are now named.

The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the 18th century.

 

之發展，直到一七一五年方可稱完整。再從歐拉

Leonhard Euler

對數學分析的貢獻，及與白努利家族之情誼講起︰

Analysis

The development of infinitesimal calculus was at the forefront of 18th century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler’s work. While some of Euler’s proofs are not acceptable by modern standards of mathematical rigour[34] (in particular his reliance on the principle of the generality of algebra), his ideas led to many great advances. Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as

e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}=\lim _{n\to \infty }\left({\frac {1}{0!}}+{\frac {x}{1!}}+{\frac {x^{2}}{2!}}+\cdots +{\frac {x^{n}}{n!}}\right).

Notably, Euler directly proved the power series expansions for e and the inverse tangent function. (Indirect proof via the inverse power series technique was given by Newton and Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the famous Basel problem in 1735 (he provided a more elaborate argument in 1741):[34]

\sum _{n=1}^{\infty }{1 \over n^{2}}=\lim _{n\to \infty }\left({\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+\cdots +{\frac {1}{n^{2}}}\right)={\frac {\pi ^{2}}{6}}.

 
A geometric interpretation of Euler’s formula

Euler introduced the use of the exponential function and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers, thus greatly expanding the scope of mathematical applications of logarithms.[32] He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions. For any real number φ (taken to be radians), Euler’s formula states that the complex exponential function satisfies

  e^{i\varphi }=\cos \varphi +i\sin \varphi .\,

A special case of the above formula is known as Euler’s identity,

  e^{i\pi }+1=0\,

called “the most remarkable formula in mathematics” by Richard P. Feynman, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, e, i and π.[35] In 1988, readers of the Mathematical Intelligencer voted it “the Most Beautiful Mathematical Formula Ever”.[36] In total, Euler was responsible for three of the top five formulae in that poll.[36]

De Moivre’s formula is a direct consequence of Euler’s formula.

In addition, Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. He also found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis. He also invented the calculus of variations including its best-known result, the Euler–Lagrange equation.

Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions. For example, he proved the infinitude of primes using the divergence of the harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler’s work in this area led to the development of the prime number theorem.[37]

 

白努利數之生成函數 \frac{x}{e^x -1} = \sum \limits_{k=0}^{\infty} B_k \frac{x^k}{k !} 或出自歐拉之手乎？否則哪來的一七三五年之歐拉-麥克勞林求和公式耶！！？？

Euler–Maclaurin formula

In mathematics, the Euler–Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, and Faulhaber’s formula for the sum of powers is an immediate consequence.

The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735 (and later generalized as Darboux’s formula). Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals.

……

The formula is often written with the subscript taking only even values, since the odd Bernoulli numbers are zero except for  {\displaystyle B_{1},} in which case we have [1][2]

{\displaystyle \sum _{i=m+1}^{n}f(i)=\int _{m}^{n}f(x)\,dx+{\frac {f(n)-f(m)}{2}}+\sum _{k=1}^{\lfloor p/2\rfloor }{\frac {B_{2k}}{(2k)!}}(f^{(2k-1)}(n)-f^{(2k-1)}(m))+R.}

───

 

誠如柯西之所言，『代數的普適性』

Generality of algebra

In the history of mathematics, the generality of algebra was a phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange,[1] particularly in manipulating infinite series. According to Koetsier,[2] the generality of algebra principle assumed, roughly, that the algebraic rules that hold for a certain class of expressions can be extended to hold more generally on a larger class of objects, even if the rules are no longer obviously valid. As a consequence, 18th century mathematicians believed that they could derive meaningful results by applying the usual rules of algebra and calculus that hold for finite expansions even when manipulating infinite expansions. In works such as Cours d’Analyse, Cauchy rejected the use of “generality of algebra” methods and sought a more rigorous foundation for mathematical analysis.

An example[2] is Euler’s derivation of the series

  {\frac {\pi -x}{2}}=\sin x+{\frac {1}{2}}\sin 2x+{\frac {1}{3}}\sin 3x+\cdots
     
 
(1)

for  0<x<\pi. He first evaluated the identity

{\frac {1-r\cos x}{1-2r\cos x+r^{2}}}=1+r\cos x+r^{2}\cos 2x+r^{3}\cos 3x+\cdots
     
 
(2)

at  r=1 to obtain

0={\frac {1}{2}}+\cos x+\cos 2x+\cos 3x+\cdots .
     
 
(3)

The infinite series on the right hand side of (3) diverges for all real  x. But nevertheless integrating this term-by-term gives (1), an identity which is known to be true by modern methods.

註︰

e^{i n \theta} = \cos(n \theta) + i \sin(n \theta)

\sum \limits_{n=0}^{\infty} {(r e^{i \theta} )}^n = 1 + r e^{i \theta} + r^2 {\left( e^{i \theta} \right) }^2 + r^3 {\left( e^{i \theta} \right)}^3 + \cdots

= 1 + r e^{i \theta} + r^2 e^{i 2 \theta} + r^3 e^{i 3 \theta} + \cdots

= \frac{1}{1 - r e^{i \theta}}

= \frac{1}{(1 - r \cos(\theta)) - i r \sin(\theta)}

= \frac{(1 - r \cos(\theta)) + i r \sin(\theta)}{{(1 - r \cos(\theta))}^2 + r^2 {\sin(\theta)}^2}

\therefore \frac{1 - r \cos(\theta)}{1 - 2 r \cos(\theta) + r^2} = 1 + r \cos(\theta) + r^2 \cos(2 \theta) + r^3 \cos(3 \theta) + \cdots

 

原理邏輯不嚴謹，但能說它不是滿富直覺與啟發性嗎？？！！

於理還是得問為什麼 (1) 能推導得到 (2) 呢？

By equating coefficients, we find that:

\sum \limits_{k = 0}^{n - 1} k^p = \frac 1 {p + 1} \sum \limits_{i = 0}^p \binom {p + 1} i B_i n^{p + 1 - i} \ \ \ \ \ (1)

\implies \ \ \sum \limits_{k = 1}^n k^p = \frac 1 {p + 1} \sum \limits_{i = 0}^p \left({-1}\right)^i \binom {p + 1} i B_i n^{p + 1 - i} \ \ \ \ \ (2) since B_1 = - \frac{1}{2} and Odd Bernoulli Numbers Vanish

 

且將 (1) 兩邊加上 n^p 。

天道左旋從左起， \sum \limits_{k = 0}^{n - 1} k^p + n^p = \sum \limits_{k = 1}^{n} k^p 。因為 0^0  定為『1』，又有 0^p = 0, \ p \ge 1。補足 n^p 式子中，假借 n^0 取代 0^0 後，等量改寫等值成。

地道右動變化生， \frac 1 {p + 1} \sum \limits_{i = 0}^p \binom {p + 1} i B_i n^{p + 1 - i} + n^p = \frac 1 {p + 1} \sum \limits_{i = 0}^p \left({-1}\right)^i \binom {p + 1} i B_i n^{p + 1 - i} 。在於除了 B_1 = - \frac{1}{2} 外， B_{2i+1} 皆為 0。偏巧 B_1 恰為 n^p 之係數，正逢 - \frac{1}{2} + 1 =\frac{1}{2} 時，因此變號 {(-1)}^i 剛好可納藏，奇偶數值正相當 ☆