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Taylor and Maclaurin Series
---------------------------
A Taylor series is a representation of a function
as an infinite sum of terms that are calculated
from the values of the function's derivatives at
a single point. It is common practice to approximate
a function by using a finite number of terms of its
Taylor series.
f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^{2}/2!
+ f'''(a)(x - a)^{3}/3!...
= Σf^{n}(x - a)^{n}/n!
^{n}
If the Taylor series is centered at zero, then
that series is also called a Maclaurin series.
f(x) = f(0) + f'(a)(x) + f''(a)(x)^{2}/2!
+ f'''(a)(x)^{3}/3!...
= Σf^{n}(x)^{n}/n!
^{n}
Ex:
cos(x) = cos(0) - xsin(0) - x^{2}/2cos(0) ...
= 1 - x^{2}/2! + x^{4}/4!
sin(x) = sin(0) + xcos(0) - x^{2}/2sin(0) ...
= x - x^{3}/3! + x^{5}/5!
e^{x} = 1 + x + x^{2}/2 + x^{3}/6 ...
Euler's formula:
e^{ix} = 1 + ix - x^{2}/2! - ix^{3}/3! ...
= (1 - x^{2}/2! ...) + i(x - x^{3}/3! ...)
= cos(x) + isin(x)
The Taylor series for 2 variables is:
f(x,y) = f(a,b) + (x - a)∂f/∂x + (y - b)∂f/∂y
+ (1/2)(x - a)^{2}∂^{2}f/∂x^{2}
+ (x - a)(y - b)∂^{2}f/∂x∂y
+ (1/2)(y - b)^{2}∂^{2}f/∂y^{2}]
f(x,y) = f(0,0) + s∂f/∂s + t∂f/∂t
+ (1/2)s^{2}∂^{2}f/∂s^{2}
+ st∂^{2}f/∂s∂t
+ (1/2)t^{2}∂^{2}f/∂t^{2}]