Difference between revisions of "Taylor series"
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<math>f(x)=f(x_0)+(x-x_0)\frac{df(x)}{dx}+\frac{(x-x_0)^2}{2!}\frac{d^2f(x)}{dx^2}+\ldots+\frac{(x-x_0)^N}{N!}\frac{d^Nf(x)}{dx^N}</math> | <math>f(x)=f(x_0)+(x-x_0)\frac{df(x)}{dx}+\frac{(x-x_0)^2}{2!}\frac{d^2f(x)}{dx^2}+\ldots+\frac{(x-x_0)^N}{N!}\frac{d^Nf(x)}{dx^N}</math> | ||
| − | where each of the derivatives is to be evaluated at <math>x=x_0</math>. If as <math>N\rightarrow\infty</math> the series converges | + | where each of the derivatives is to be evaluated at <math>x=x_0</math>. If as <math>N\rightarrow\infty</math> the series converges, then it is exact. Otherwise, it can be used as an approximation. Often, Taylor series are performed around <math>x_0=0</math>, in which case they are sometimes also known as a [[Maclaurin series]]. |
===Examples of common Taylor series=== | ===Examples of common Taylor series=== | ||
Revision as of 17:22, June 3, 2008
The Taylor series of a function is useful for approximating a mathematical function near to some particular point. For a function , the Taylor series about the point is
where each of the derivatives is to be evaluated at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=x_0} . If as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\rightarrow\infty} the series converges, then it is exact. Otherwise, it can be used as an approximation. Often, Taylor series are performed around Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0=0} , in which case they are sometimes also known as a Maclaurin series.
Examples of common Taylor series
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots}
Extensions of the Exponential Function
Consider the exponential of imaginary number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle yi} ,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{yi}=1+yi+\frac{(yi)^2}{2!}+\frac{(yi)^3}{3!}+\frac{(yi)^4}{4!}+\frac{(yi)^5}{5!}+\ldots}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =1+yi-\frac{y^2}{2!}-i\frac{y^{3}}{3!}+\frac{y^4}{4!}+i\frac{y^5}{5!}-\ldots}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =(1-\frac{y^2}{2!}+\frac{y^4}{4!}-\ldots)+i(y-\frac{y^{3}}{3!}+\frac{y^5}{5!}-\ldots)}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\cos y+i\sin y}
By the power laws then all complex numbers have an exponential,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{x+yi}=e^{x}(\cos y+i\sin y)} .