Euler’s method of construction of the Exponential function

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1 Euler’s method of construction of the Exponential function
北京景山学校 • 纪光老师 • Dec.2010 北京景山学校 － 激光老师－ Dec.2009

2 Objective : build a function defined on such that for any , f’(x) = f(x) and f(0) = 1
北京景山学校 • 纪光老师 • Dec.2010

3 Objective : build a function defined on such that for any , f’(x) = f(x) and f(0) = 1
Reminder : from the definition of the derivative of a function f in one point a : 北京景山学校 • 纪光老师 • Dec.2010

4 Objective : build a function defined on such that for any , f’(x) = f(x) and f(0) = 1
Reminder : from the definition of the derivative of a function f in one point a : 北京景山学校 • 纪光老师 • Dec.2010

5 Objective : build a function defined on such that for any , f’(x) = f(x) and f(0) = 1
Reminder : from the definition of the derivative of a function f in one point a : f(a + h) = f(a) +h.f’(a)+ 北京景山学校 • 纪光老师 • Dec.2010

6 then f(a + h) ≈ f(a) +h.f’(a) +……
For any small value of h, close to 0, then f(a + h) ≈ f(a) +h.f’(a) +…… 北京景山学校 • 纪光老师 • Dec.2010

7 For any small value of h, close to 0,
then f(a + h) ≈ f(a) +h.f’(a) +…… In this case we have f’(a) = f(a) and f(0) = 1 then for a =0  f(0 + h) ≈ f(0) +h.f’(0) = f(0) +h.f(0) =1 +h f(h) ≈ 1 + h 北京景山学校 • 纪光老师 • Dec.2010

8 f(h) ≈ 1 + h f(2h) ≈ f(h). (1+h) = (1+ h)2
For any small value of h, close to 0, then f(a + h) ≈ f(a) +h.f’(a) +…… In this case we have f’(a) = f(a) and f(0) = 1 then for a =0  f(0 + h) ≈ f(0) +h.f’(0) = f(0) +h.f(0) =1 +h f(h) ≈ 1 + h  f(2h) = f(h +h) ≈ f(h) + h.f’(h) f(2h) ≈ f(h). (1+h) = (1+ h)2 北京景山学校 • 纪光老师 • Dec.2010

9 Let un = f(nh) un+1 = un(1+h) un = f(nh) ≈ (1 + h)n
f[(n+1)h] =f(nh +h) = f(nh)(1 +h) un+1 = un(1+h) The sequence (un) is Geometric 1st term u0 =f(0) = 1 reason q =1+h un = f(nh) ≈ (1 + h)n 北京景山学校 • 纪光老师 • Dec.2010

10 f(nh) ≈ (1 + h)n for example let and n = 100 then
北京景山学校 • 纪光老师 • Dec.2010

11 f(nh) ≈ (1 + h)n for example let and n = 100 then then for any Integer k
北京景山学校 • 纪光老师 • Dec.2010

12 北京景山学校 • 纪光老师 • Dec.2010

13 then for n = 100 we have : and for any Integer k we have : f(k) = ek
北京景山学校 • 纪光老师 • Dec.2010

14 then for n = 100 we have : and for any Integer k we have : f(k) = ek
then for n = 100 we have : and for any Integer k we have : f(k) = ek Question : if x is any Real number can we write f(x) = ex ??? 北京景山学校 • 纪光老师 • Dec.2010

15 If n is a very large number compared to x, then we could let
北京景山学校 • 纪光老师 • Dec.2010

16 Problem : the formula : is “wrong” …
If n is a very large number compared to x, then we could let Problem : the formula : is “wrong” … 北京景山学校 • 纪光老师 • Dec.2010

17 because it was established that
The formula : is “wrong” … because it was established that only for the Integers m and n … …not for Real numbers … yet !) 北京景山学校 • 纪光老师 • Dec.2010

18 Let’s look at the present situation more closely with the computer…
So what ? ! ? Let’s look at the present situation more closely with the computer… (Euler had a fabulous capacity of calculations and did not need to use a computer …) 北京景山学校 • 纪光老师 • Dec.2010

19 北京景山学校 • 纪光老师 • Dec.2010

20 In observing the spread sheet, we can see that for any real number x written in decimal form we can find a number N such that : We may decide to write Exp(0.67) or e0.67to match the previous results for the integers. 北京景山学校 • 纪光老师 • Dec.2010

21 (since it has a derivative).
So what ? ! ? … Euler was able to calculate these values with a great acuracy, but he was mainly able to prove that the function defined by the original conditions f’(x) = f(x) and f(0) =1 is defined for any real and is continuous. (since it has a derivative). 北京景山学校 • 纪光老师 • Dec.2010

22 So what ? ! ? … Euler was able to calculate these values with a great acuracy, but he was mainly able to prove that the function defined by the original conditions f’(x) = f(x) and f(0) =1 is defined for any real number and is continuous. (since it has a derivative). He named it “Exponential” and he was also able to prove that this new function Exp. had a the fundamental property of transforming sums into products for any real numbers u and v Exp (u + v) = Exp(u) • Exp(v) Or eu+v = eu. ev Then the previous formula makes sense. 北京景山学校 • 纪光老师 • Dec.2010

23 Demo of the fundamental formula of the Exponential function (1)
Let’s prove that if f’(x) = f(x) and f(0) = 1 then for any real numbers u and v we can write f(u + v) = f(u).f(v) 北京景山学校 • 纪光老师 • Dec.2010

24 Demo of the fundamental formula of the Exponential function (2)
Let F(x) =f(a+x).f(-x) Then F’(x) =f’(a+x).f(-x) + f(a+x).[-f’(-x)] = f(a+x)•[f(-x)-f(-x)] = 0 Then F(x) is a constant : F(x)=F(0)=f(a) => f(x+a).f(-x) = f(a) And particularly for a =0 f(x).f(-x) = f(0) =1 北京景山学校 • 纪光老师 • Dec.2010

25 Demo of the fundamental formula of the Exponential function (3)
Then by multiplying both members by f(x) in the previous equation we have : f(a+x).f(-x).f(x)=f(a).f(x) hence : f(a+x) = f(a).f(x) Or f(u + v) = f(u).f(v) 北京景山学校 • 纪光老师 • Dec.2010

26 Demo of the fundamental formula of the Exponential function (4)
Then if we use the exponential notation : Exp(u +v) = Exp(u).Exp(v) eu + v = eu.ev 北京景山学校 • 纪光老师 • Dec.2010

27 More over Euler established a fundamental formula of development of functions in “power series” such that : 北京景山学校 • 纪光老师 • Dec.2010

28 More over Euler established a fundamental formula of development of functions in “power series” such that : It is this formula (using at least 6 or 7 terms) that is used in computers and pocket calculators to provide us with the “exact” values of Exp(x). Then we can check that the values found by the differential relationship are correct approximate values. 北京景山学校 • 纪光老师 • Dec.2010

29 Next week we will prove that the reciproqual is true : If a function f is not null and has a derivative on and if for any numbers u and v we have the formula f(u + v) = f(u).f(v) Then : f’(x) = f(x) 北京景山学校 • 纪光老师 • Dec.2010

30 谢谢 北京景山学校 • 纪光老师 • Dec.2010