Exponential function: Difference between revisions

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The '''exponential function''' is one of the most important [[function]]s in [[mathematics]]. It is written as exp(''x'') or ''e''^''x'' (where ''e'' is the [[e - base of natural logarithm|base of the natural logarithm]]) and can be defined in two equivalent ways:

∞ ''x''^''n''

exp(''x'') = ∑ ---

''n''=0 ''n''!

exp(''x'') = lim (1 + ''x''/''n'')^''n''

''n''→∞

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

If ''x'' is real, then exp(''x'') is positive and strictly increasing. Therefore its [[inverse function]], the [[natural logarithm]] ln(''x''), is defined for all positive ''x''. Using the natural logarithm, one can define more general exponential functions as follows:

''a''^''x'' = exp(ln(''a'') ''x'')

for all ''a'' > 0 and all real ''x''.

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

Exponential functions "translate between addition and multiplication" as is expressed in the following ''exponential laws'':

''a''⁰ = 1

''a''^{''x'' + ''y''} = ''a''^''x'' ''a''^''y''

''a''^(''xy'') = (''a''^''x'')^''y''

''a''^''x'' ''b''^''x'' = (''ab'')^''x''

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 [[derivative]]s. If a variable's growth or decay rate is proportional to its size, as is the case in unlimited population growth, continuously compounded interest or radioactive decay, then the variable can be written as a constant times an exponential function of time.

When considered as a function defined on the complex plane, the exponential function retains the properties

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

d/d''x'' ''a''^''bx'' = ln(''a'') ''b'' ''a''^''bx''.

The major importance of the exponential functions in the sciences stems from the fact that they are constant multiples of their own [[derivative]]s. If a variable's growth or decay rate is proportional to its size, as is the case in unlimited population growth, continuously compounded interest or radioactive decay, then the variable can be written as a constant times an exponential function of time.

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

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

for all complex numbers (or even Banach algebra elements) ''z'' and ''w''. The exponential function

for all complex numbers (or even Banach algebra elements) ''z'' and ''w''. The exponential function on the complex plane

is periodic with imaginary period 2π''i'', and this is the reason that extending the natural logarithm to complex arguments naturally yields a multi-valued function ln(''z''). We can define a more general exponentiation:

''z''^''w'' = exp(ln(''z'') ''w'')

for all complex numbers ''z'' and ''w''.

This is then also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued functions.

In the context of general non-commutative Banach algebras, such as algebras of matrices or operators on [[Banach space|Banach]] or [[Hilbert space|Hilbert]] spaces, the exponential function is typically considered to be a function of a real argument:

''f''(''t'') = exp(''t'' ''A'')

where ''A'' is a fixed element of the algebra and ''t'' is any real number. This function has the important properties

''f''(''s'' + ''t'') = ''f''(''s'') ''f''(''t'')

d/d''t'' ''f''(''t'') = ''A'' ''f''(''t'')

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

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/Talk