Exponential function: Difference between revisions

Revision as of 23:44, 16 January 2003

Revision as of 06:49, 19 February 2003

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

: <math>{dy \over dx} = y</math>

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'sSchroedinger equation|Schrödinger equation]] and the [[Laplace's equation]] as well as the equations for [[simple harmonic motion]].

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

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: <math>\exp(z + w) = \exp(z) \exp(w)</math>

: <math>\exp(0) = 1</math>

: <math>\exp(''z'') \neq&ne 0</math>

: <math>\exp'&prime(''z'') = \exp(''z'')

for all ''z'' and ''w''. The exponential function on the complex plane is a [[holomorphic function]] which is periodic with imaginary period <math>2 \pi i</math> which can be written as