The discovery of e began with differentiation. When h gradually approaches zero, the calculated value is infinitely close to a certain value of 2.7 1828 ..., which was first discovered by the famous Swiss mathematician E. Euler. He named the irrational number after himself, prefixed with lowercase e.

Calculate the derivative of logarithmic function, and get that when a=e, the derivative of is 0, so the logarithm based on e is more reasonable, which is called natural logarithm.

If the exponential function ex is Taylor expansion, then

Substitute x= 1 into the above formula.

This series converges quickly, and the value of E is approximately 40 decimal places.

When the exponential function ex is extended to the complex number z=x+yi, it is determined by the following formula.

Through this series of calculations, we can get

From this, Demol's theorem, the formula of sum and difference angles of trigonometric functions and so on can be easily deduced. For example, z 1 = x 1+y 1i, z2 = x2+y2i,

On the other hand,

So,

We can not only prove that e is an irrational number, but also a transcendental number, that is, it is not the root of any integer coefficient polynomial. This result was obtained by Hermite in 1873.

A) differences.

Consider a discrete function (sequence) R, whose value u(n) at n is denoted as un, and we usually write this function as OR (un). The difference of the sequence U is still a sequence, and its value in n is defined as

In the future, let's remember Jane

(Example): The difference sequence of sequence 1, 4, 8, 7, 6, -2, ... is 3, 4,-1, -8. ...

Note: We say "series" is "a function defined at discrete points". This statement sucks in high school, but it is appropriate here because it has a completely parallel analogy with continuous functions.

The nature of difference operator

(a) [collectively referred to as linear]

(ii) (Constant) [Basic Theorem of Difference Equation]

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Where (n(k) is called a permutation sequence.

(4) called natural geometric series.

(iv) The difference sequence (i.e. "derivative function") of the' general exponential sequence (geometric sequence) rn is rn(r- 1).

(2). Sum integral

Give a series of numbers (un). The problem of summation is to calculate summation. How to calculate? We got the following important results:

Theorem 1 (basic theorem of difference and division) If we can find a sequence (vn), then

Sum and fraction also have linear properties:

A) differences

Given a function f, if the limit of Newton's quotient (or difference quotient) exists, then we call this limit value f the derivative of point x0 and write it as f'(x0) or Df(x), that is.

If the derivative of f exists at every point in the defined area, it is called a differentiable function. We call it the derivative function of f, not the differential operator.

Properties of differential operators:

(a) [collectively referred to as linear]

(ii) (Constant) [Basic Theorem of Difference Equation]

(3) Dxn=nxn- 1

Dex = ex

(iv)' The derivative function of the general exponential sequence ax is

(b) integration.

Let f be a function defined on [a, b], and the problem of integration is to calculate the shadow area of Figure A. Our method is to divide [a, b]:

; Secondly, take a sample point [xi- 1, xi] for each small segment; Find the approximate sum again (see Figure B); Finally, take the limit (let the length of each segment approach 0).

If this limit exists, we will remember that the geometric meaning is the area of shadow in Figure A. 。

(In fact, continuity is also "almost" a necessary condition for the existence of integral. )

Tujia nationality

Figure b

Integral operators also have linear properties:

Theorem 2 If F is a continuous function, it exists. (In fact, continuity is also a necessary condition for the existence of an integral. )

Theorem 3 (Basic Theorem of Calculus) Let f be a continuous function defined in the closed interval [a, b], and we want to get the integral. If we can find another function g that makes g'=f, then

Note: (1)(2) Although the two formulas are analogies, there is one difference, that is, be careful about the upper limit of the sum!

Theorem 1 and Theorem 3 above basically say that difference sum, differential and integral are two reciprocal operations, just as addition, subtraction, multiplication and division are reciprocal operations.

As we all know, the operation of difference and differential is much simpler than the operation of sum, fraction and integral. Theorem 1 and Theorem 3 above tell us that to calculate the sum of (un) and the integral of fraction and f, we only need to find another (vn) and g to satisfy G'= F (this is a problem of difference and differentiation), and then we can get the answer by substituting vn and g into the upper and lower limits. In other words, we can use something simpler.

A) Taylor expansion formula

There are discrete and continuous analogies. It is a special case of the important idea of approximation in mathematics. The idea of approximation is this: given a function F, we should study its behavior, but F itself may be very complicated and difficult to handle, so we try to find a simpler function G to make it "close" to F, and then we use G instead of F, which is to simplify the complexity.

Two questions: How to choose simple function and approximate scale?

(1) In the case of continuous world, the approximation idea of Taylor expansion is to select polynomial function as simple function and local tangency as approximation scale. More specifically, given a function f that can be differentiated to order n, we need to find a polynomial function g of order n so that it is "tangent" to f at point x0, that is, the answer is

This formula is called the n-order Taylor expansion of f at point x0.

G is very close to F near x0, so we use G to replace F locally, so we can use G to find some local qualitative behaviors of F, so Taylor expansion is only a local approximation. When F is a good enough function, that is, the so-called analytic function, F can be expanded into Taylor series, which is equal to F itself.

It is worth noting that in the special case of first-order Taylor expansion, the graph of g(x)=f(x0+f'(x0)(x-x0)) is just a straight line that passes through the graph of points (x0, f(x0)) and is tangent to F. Therefore, the significance of the first-order Taylor expansion of f at point x0 is that we have used the point (x0).

Talor expansion can help us to do many things, such as judging the maximum and minimum values of functions, finding the approximate value of integrals, and making function tables (such as trigonometric function tables and logarithmic tables). In fact, we can use approximate ideas to "consistently" calculus.

Many times, we notice that we choose polynomial function as simple approximation function for a simple reason: among many elementary functions, such as trigonometric function, exponential function, logarithmic function, polynomial function and so on. From the arithmetic point of view, polynomial function is the simplest, because to calculate the value of polynomial function, it only involves four operations of addition, subtraction, multiplication and division, and other functions are not so simple.

Of course, from other analysis angles, in some cases, there are other simple functions that are more useful and important. For example, trigonometric polynomials, combined with some approximation scales, give us Fourier series expansion, which plays an important role in applied mathematics. (In fact, Fourier series expansion is an approximation scale with the smallest variance, which often appears in higher mathematics and is also applied in statistics. )

Note: Take the special case of x0=0 as an example. At this time, Taylor expansion is also called Ma Kraulin expansion. However, as long as we can expand a special case and want a general Taylor expansion, it is good to do translation (or variable substitution). Therefore, Taylor expansion can only be done at the point of x=0 from the beginning.

(2) For discrete cases, Taylor expansion is:

Given a sequence, we need to find a polynomial sequence with degree n (gt) so that gt and ft have N-order "difference approximation" when t=0. The so-called zero-order n-order difference approximation refers to:

The answer is that this formula is Maclaurin formula in discrete case.

B) analogy between partial integral formula and Abel partial sum formula

(a) partial integral formula:

Let u (x) and v (x) be continuous on [a, b], then

(2) Abel partial sum formula:

Let (UN) and (v) be two series, so Sn = U 1+...+UN, then

The above two formulas are Leibniz's derivative formula D(uv)=(Du)v+u(Dv) and Leibniz's difference formula respectively. Note that one of the two Leibniz formulas is very symmetrical, while the other is asymmetrical.

(d) compound interest and continuous compound interest (this is also an analogy between discrete and continuous respectively)

(1) The compound interest problem is as follows: there is principal y0, annual interest rate R, and compound interest once a year. Ask the principal and interest after N years and yn= Obviously, this series satisfies the difference equation yn+ 1=yn( 1+r).

According to (2) of (c), we know that yn=y0( 1+r)n is a compound interest formula.

(2) If compound interest is considered m times a year, the sum of principal and interest after t years is

Order, you get the concept of continuous compound interest, and the sum of principal and interest at this time is y(t)= yoert.

In other words, the principal and interest of time t and y(t)= yoert are the solutions of the differential equation y'=ry.

As can be seen from the above, the discrete compound interest problem is described by difference equation, while the continuous compound interest problem is described by differential equation. For linear difference equations and differential equations with constant coefficients, the whole point of solving equations is the superposition principle, so the solving methods are completely parallel.

(e) Fubini's theorem of multiple sums and points and Fubini's theorem of multiple integrals (also an analogy between discrete and continuous).

(1) Fubini's double sum theorem: given a series of numbers (ars) with double exponents, we want to sum r= 1 to m and S = 1 to n (ars), then this sum can be obtained as follows: light sum r, and then sum s (and vice versa). In other words, we have.

(2) Fubini's multiple integral theorem: Let f(x, y) be an integrable function defined on, then

Of course, several variables are the same.

The concept of leberg integral

(1) Discrete case: Given a series (an), the sum should be estimated. Leberg's idea is that no matter what the order of the data indicators of this pile is, we only divide it into piles according to the size of the values, and then multiply a value from each pile by the number of the piles to get the overall sum.

(2) Continuity: Given a function f, we need to define the area enclosed by the curve y=f(x) and the X axis from A to B (see the figure below).

Leberg's idea is to divide the shadow domain of F:

X whose function value is between yi- 1 and yi is set together to make it 0, thus [a, b] is divided into sampling points and approximately summed.

If the limit of the above approximate sum exists, it is called Lebesgue integral of f on [a, b].

In mathematics, e is one of the most commonly used transcendental numbers.

It is usually used as the base of natural logarithm, that is, In(x)= logarithm (1) series of x with base e or function f(n)=( 1+ 1/n)n when n→∞ =e or g (n) = (1) ), n is a natural numb from 0 to infinity. That is1+11! + 1/2! + 1/3! +……( 1-2)e^x=sum(( 1/n！ )x n)( 1-3)[n n/(n- 1)(n- 1)]-[(n- 1)(n- 1)/(n Cosx =(e IX+e(-IX))/2 = re(e IX)， Isinx = = (e ix-e (-ix))/2 = iim (e ix) So we can combine the simple properties of trigonometric functions or hyperbolas to calculate relatively complicated formulas, such as the sum angle (2- 1) e x = coshx+sinhx, which means hypcosx+hypsinx, and is also recorded as CHX and SHX. Then click1hypsin+(1hypcos1) or enter 1hs+( 1ho)= or (1hs+( 1ho)). Simply put, you can click 1 inv Ln or enter 1in, which is actually to calculate E 1. We can also get: e = 2.7182818284 59045 23536 02874 713526 (the decimal place of 31is rounded to 7) (4) This is two thousand digits after the decimal point: e = 2.7. 4 7 1352 66249 77572 47093 69995 95749 66967 62772 40766 30353 54759 457 13 82 178 525 16 64274 27466 39 1 93 20030 5992 1 8 174 1 35966 29043 57290 03342 95260 59563 0738 1 32328 62794 34907 63233 82988 0753 1 9525 / kloc-0/ 0 190 1 15738 34 187 93070 2 1540 89 149 93488 4 1675 09244 76 146 06680 82264 800 / kloc-0/6 84774 1 1853 74234 54424 37 107 5 3907 77449 92069 55 170 276 18 38606 26 133 1 3845 83000 75204 49338 26560 29760 6737 1 13200 70932 8709 1 27443 74 704 72306 96977 2093 1 0 14 1 6 92836 8 1902 55 15 1 08657 46377 2 1 1 1 2 52389 78442 50569 53696 77078 544 99 69967 94686 44549 05987 93 163 68892 30098 793 12 7736 1 782 15 42499 92295 7635 / kloc-0/ 48220 82698 95 193 66803 3 182 5 28869 39849 6465 1 05820 93923 98294 88793 32036 25094 43 1 17 30 1 23 8 1970 684 16 14039 70 198 37679 32068 32823 76464 80429 53 1 18 02328 78250 98 194 558 1 5 30 175 67 173 6 1332 0698 1 12509 96 18 1 88 159 304 16 9035 1 59888 85 / kloc-0/93 45807 27386 67385 89422 87922 84998 92086 80582 57492 796 10 484 19 84443 63463 24496 8 4875 60233 62482 704 1 9 78623 20900 2 1609 90235 30436 994 18 49 146 3 1409 343 17 38 143 64054 6253 1 52096 65 438+08369 08887 070 16 76839 64243 78 140 5927 1 45635 4906 1 303 10 72085 10383 7505 1 0 1 157 47704 654 38+07 189 86 106 87396 96552 1267 1 54688 95703 50354 02 123 40784 98 1 93 3432 1 068 17 0 12 10 0562 7 88023 5 1930 33224 7450 1 58539 04730 4 1995 77770 93503 6604 / kloc-0/ 69973 29725 08868 76966 40355 5707 1 62268 447 16 25607 98826 5 1787 134 19 5 1246 6520 1 03059 2 1236 677 19 43252 78675 39855 89448 96970 96409 75459 18569 56380 23637 0 162 1 1 2047 74272 28364 896 13 4225 1 64450 78 182 44235 29486 36372 14 174 02388 9344 1 24796 3 5743 70263 75529 44483 37998 0 16 12 54922 78509 25778 25620 92622 64832 62779 33386 56648 16277 25 164 0 / kloc-0/9 10 59004 96 5438+0644 99828 93 150 56604 72580 27786 3 1864 155 19 56532 44258 69829 46959 3080 1 9 1529 872 1 1 72556 347 54 63964 479 10 14590 40905 86298 49679 12874 06870 50489 58586 7 1 747 98546 67757 57320 568 12 88459 2054 1 33405 3922 0 00 1 13 78630 09455 60688 16674 00 1 69 84205 58040 33637 95376 45203 04024 32256 6 1352 78369 5 1 177 88386 38744 39662 53224 98506 54995 88623 428 1 8 99707 73327 6 17 17 83928 03494 650 14 34558 89707 19425 86398 77275 47 109 62953 74 152 1 / kloc-0/ 15 1 36835 06275 26023 26484 72870 39207 643 10 05958 4 1 166 12054 52970 30236 47254 92966 6 938 / kloc-0/ 15 137 32275 36450 98889 03 136 02057 248 17 6585 1 18063 03644 28 123 1 4965 50704 75 102 54 465 0 1 172 72 1 15 55 194 86685 08003 68532 28 183 152 1 9 60037 35625 27944 95 158 284 18 82947 876 10 85263 98 139