Function:
We define the exponential function as (the only non trivial function) which is it’s
own derivative:
Derivative of the exponential function:
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Just using the definition of the factorial function we find the Taylor series expansion of the exponential function
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First we will prove some very important properties of the exponential function.
Fundamental addition formula of the exponential function:
Applying
the definition of the e-function as a series we find
Euler’s formula:
taking into account
This is a very important relation. It can be
understood as the definition of the sin and cos function and allows to replace
For real numbers
Additionally we directly get
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Addition theorems for sin and cos functions:
Combining the exp-addition
formula with Euler’s formula we find
From
real and imaginary part we finally get (representing the even and odd part of the complex exponential function)
Combining both equations we easily get
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Back to complex numbers:
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Example:
Has the equation
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Similar to the definition of the cos and sin function we have
Definition 7
hyperbolic functions
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Theorem for
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Definition 8
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e.g.:
The above
relation between sin, cos, sinh, and cosh allow e.g. to easily apply the addition theorems to calculate
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© J. Carstensen (Math for MS)