Exponential function: Difference between revisions

Revision as of 19:32, 6 July 2002

Revision as of 21:34, 6 July 2002

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:√ ''a'' = ''a''^1/2

:^''n''√ ''a'' = ''a''^1/''n''

=== Exponential function and differential equations ===

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

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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.

The exponential function thus solves the basic [[differential equation]]

dy

-- = y

dx

and it is for this reason commonly encountered in differential equations. In particular

the solution of linear ordinary [[differential equation]]s

can frequently be written in terms of exponential functions.

These equations include [[Schrodinger's equation]] and the

[[Laplace equation]] as well as the equations for [[simple harmonic motion]].

=== Exponential function on the complex plane ===

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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.

=== Exponential function and differential equations ===

The exponential function solves the basic equation

dy

-- = y

dx

and for this reason is exponential function is frequently

encountered in differential equations. In particular

the solution of linear ordinary [[differential equation]]s

can frequently be written in terms of exponential functions.

These equations include [[Schrodinger's equation]] and the

[[Laplace equation]] as well as the equations for [[simple harmonic motion]].

=== Exponential function for matrices and Banach algebras ===