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public:mathematica_common_expressions [2016/10/10 12:47] Fufu Fang [Taylor Series] |
public:mathematica_common_expressions [2018/03/31 01:38] (current) |
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+ | ====== Mathematica Common Expressions ====== | ||

+ | ===== Taylor Series ===== | ||

+ | The $n^th$ term of the Taylor series expansion for the function $f(x)$ at $a$ can be obtained by the following function: | ||

+ | <code mathematica> | ||

+ | TaylorTerm[fx_, x_, a_, n_] := (((Evaluate[D[fx, {x, n}]]) /. x -> a)/n!) (x - a)^n | ||

+ | </code> | ||

+ | The sum of the first n terms for the Taylor series expansion for the function $f(x) at $a$ can be obtained by the following function: | ||

+ | <code mathematica> | ||

+ | TaylorSeries[fx_, x_, a_, n_] := Evaluate[Normal[Series[fx[y], {y, a, n}]]] /. y -> x | ||

+ | </code> | ||

+ | Note that you may have to use a pure function as the first argument, e.g. | ||

+ | <code mathematica> | ||

+ | In[1]:= TaylorSeries[Function[x, Sqrt[x + 1]], x, 0, 10] | ||

+ | Out[1]= 1 + x/2 - x^2/8 + x^3/16 - (5 x^4)/128 + (7 x^5)/256 - ( | ||

+ | 21 x^6)/1024 + (33 x^7)/2048 - (429 x^8)/32768 + (715 x^9)/65536 - ( | ||

+ | 2431 x^10)/262144 | ||

+ | </code> |