 # Euler’s method of construction of the Exponential function

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Euler’s method of construction of the Exponential function

Objective : build a function defined on such that for any , f’(x) = f(x) and f(0) = 1

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

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

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

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

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

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

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

f(nh) ≈ (1 + h)n for example let and n = 100 then

f(nh) ≈ (1 + h)n for example let and n = 100 then then for any Integer k

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
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

If n is a very large number compared to x, then we could let

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

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

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

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

(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

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

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

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

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

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

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

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

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