History of Exponential function

	Revision 10 . . November 28, 2001 2:19 am by AxelBoldt [p-adics, holomorphic, never equal to zero, more exponential laws]
	Revision 9 . . September 1, 2001 8:06 pm by AxelBoldt [Correct statement about exponential maps on Banach algebras; mention exponential map on Lie algebras]
	Revision 7 . . (edit) August 18, 2001 11:59 am by (logged).176.164.xxx [*Added Talk page]

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Difference (from prior major revision) (author diff)

Changed: 4c4

n=0 n!

n=0 n!

Changed: 9c9

(see limit and infinite series). Here n! stands for the factorial of n and x can be any real or complex number, or even any element of a Banach algebra.

(see limit and infinite series). Here n! stands for the factorial of n and x can be any real or complex number, or even any element of a Banach algebra or the field of p-adic numbers.

Changed: 13c13

ax = exp(ln(a) x)

:ax = exp(ln(a) x)

Changed: 17c17

The exponential function also gives rise to the [trigonometric functions]? (as can be seen from Euler's formula) and to the [hyperbolic functions]?. Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another.

The exponential function also gives rise to the [trigonometric functions]? (as can be seen from Euler's formula) and to the [hyperbolic functions]?. Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another.

Changed: 21,24c21,31

a0 = 1 ax + y = ax ay a(xy) = (ax)y ax bx = (ab)x

:a0 = 1 :a1 = a :ax + y = ax ay :a(xy) = (ax)y :1 / ax = (1/a)x = a-x :ax bx = (ab)x These are valid for all positive real numbers a and b and all real numbers x. Expressions involving fractions and roots can often be simplified using exponential notation because :1 / a = a-1 :√ a = a1/2 :n√ a = a1/n

Changed: 26c33

These are valid for all positive real numbers a and b and all real numbers x.

The major importance of the exponential functions in the sciences stems from the fact that they are constant multiples of their own derivatives:

Changed: 28,30c35

The major importance of the exponential functions in the sciences stems from the fact that they are constant multiples of their own derivatives: d/dx abx = ln(a) b abx.

:d/dx abx = ln(a) b abx.

Changed: 34,38c39

When considered as a function defined on the complex plane (or even on a commutative Banach algebra), the exponential function retains the properties exp'(z) = exp(z) and

When considered as a function defined on the complex plane (or even on a commutative Banach algebra or the p-adic numbers), the exponential function retains the important properties

Changed: 40c41,44

exp(z + w) = exp(z) exp(w)

:exp(z + w) = exp(z) exp(w) :exp(0) = 1 :exp(z) ≠ 0 :exp'(z) = exp(z)

Changed: 42c46

for all z and w. The exponential function on the complex plane

for all z and w. The exponential function on the complex plane is a holomorphic function which

Changed: 45c49

zw = exp(ln(z) w)

:zw = exp(ln(z) w)

Changed: 52c56

f(t) = exp(t A)

:f(t) = exp(t A)

Changed: 56,57c60,62

f(s + t) = f(s) f(t) d/dt f(t) = A f(t)

:f(s + t) = f(s) f(t) :f(0) = 1 :d/dt f(t) = A f(t)

Changed: 59c64

The "exponential map" sending a Lie algebra to the Lie group that gave rise to it has the same properties, which explains the terminology.

The "exponential map" sending a Lie algebra to the Lie group that gave rise to it shares these properties, which explains the terminology.