| (One intermediate revision by one other user not shown) | |||
| Line 3: | Line 3: | ||
! style="background-color: rgb(228, 188, 126); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Table of Taylor Series | ! style="background-color: rgb(228, 188, 126); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Table of Taylor Series | ||
|- | |- | ||
| − | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | | + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Taylor series of functions of one variable |
|- | |- | ||
| − | |<math> | + | |<math> f(x) \ = \ f(a) \ + \ f'(a)(x-a) \ + \ \frac{f''(a)(x-a)^2}{2!} \ + \ \cdot \cdot \cdot \ + \ \frac{f^{(n-1)} (a)(x-a)^{n-1}}{(n-1)!} \ + \ R_n </math> |
|- | |- | ||
| + | | Rn is the rest of the first n terms, and can be placed in one of two forms: | ||
| + | |- | ||
| + | |<math> \text{ Rest of Lagrange} \qquad R_n \ = \ \frac{f^{(n)}(\zeta)(x-a)^n}{n!}</math> | ||
| + | |- | ||
| + | |<math> \text{ Rest of Cauchy} \qquad R_n \ = \ \frac{f^{(n)}(\zeta)(x-\zeta)^{n-1}(x-a)}{(n-1)!}</math> | ||
| + | |- | ||
| + | |<math> \text{z value, which may be different in two residues, located between a and x. the result is valid if f(x) has continuous derivatives at least up to order n } </math> | ||
| + | |- | ||
| + | |<math> \text {if } \lim_{n \to \infty}R_n \ = \ 0,\ \text{ the infinite series obtained is called the taylor series of f(x) near x = a }</math> | ||
| + | |- | ||
| + | | <math> \text { if a =0 , it is often called Mac Laurin series }</math> | ||
| + | |- | ||
| + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Binomial Series | ||
| + | |- | ||
| + | |<math></math> | ||
| + | |- | ||
| + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Expansion Series of Exponential functions and logarithms | ||
| + | |- | ||
| + | |<math></math> | ||
| + | |- | ||
| + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Expansion Series of Circular Functions | ||
| + | |- | ||
| + | |<math></math> | ||
| + | |- | ||
| + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Expansion Series of Hyperbolic Functions | ||
| + | |- | ||
| + | |<math></math> | ||
| + | |- | ||
| + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Various Series | ||
| + | |- | ||
| + | |<math></math> | ||
| + | |- | ||
| + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | series of reciprocal power series | ||
| − | + | |- | |
| + | |<math></math> | ||
| + | |- | ||
| + | ! style="background-color: rgb(238, 238, 238); background-image: none; background-repeat: repeat; background-attachment: scroll; background-position: 0% 0%; -moz-background-size: auto auto; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial; font-size: 110%;" colspan="2" | Taylor Series of Functions of two variables | ||
| + | |- | ||
| + | |<math></math> | ||
| + | |- | ||
|} | |} | ||
| + | |||
---- | ---- | ||
Latest revision as of 12:42, 22 November 2010
| Table of Taylor Series | |
|---|---|
| Taylor series of functions of one variable | |
| $ f(x) \ = \ f(a) \ + \ f'(a)(x-a) \ + \ \frac{f''(a)(x-a)^2}{2!} \ + \ \cdot \cdot \cdot \ + \ \frac{f^{(n-1)} (a)(x-a)^{n-1}}{(n-1)!} \ + \ R_n $ | |
| Rn is the rest of the first n terms, and can be placed in one of two forms: | |
| $ \text{ Rest of Lagrange} \qquad R_n \ = \ \frac{f^{(n)}(\zeta)(x-a)^n}{n!} $ | |
| $ \text{ Rest of Cauchy} \qquad R_n \ = \ \frac{f^{(n)}(\zeta)(x-\zeta)^{n-1}(x-a)}{(n-1)!} $ | |
| $ \text{z value, which may be different in two residues, located between a and x. the result is valid if f(x) has continuous derivatives at least up to order n } $ | |
| $ \text {if } \lim_{n \to \infty}R_n \ = \ 0,\ \text{ the infinite series obtained is called the taylor series of f(x) near x = a } $ | |
| $ \text { if a =0 , it is often called Mac Laurin series } $ | |
| Binomial Series | |
| Expansion Series of Exponential functions and logarithms | |
| Expansion Series of Circular Functions | |
| Expansion Series of Hyperbolic Functions | |
| Various Series | |
| series of reciprocal power series | |
| Taylor Series of Functions of two variables | |
