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[[Category:Formulas]]
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keywords: Taylor, Geometric, Binomial
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<center><font size= 4>
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'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
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</font size>
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 +
'''Power Series'''
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(Used in [[ECE301]] and [[ECE438]])
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 +
</center>
 +
 +
----
 +
 
{|
 
{|
 
|-  
 
|-  
−
! colspan="2" style="background:  #e4bc7e; font-size: 110%;" | Power Series Formulas
+
! colspan="2" style="background:  #e4bc7e; font-size: 110%;" | [[Taylor_maclaurin_series|Taylor Series]] Formulas
 
|-
 
|-
−
! colspan="2" style="background: #eee;" | Geometric Series
+
! colspan="2" style="background: #eee;" | Series in symbolic forms
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | Finite Geometric Series Formula || <math>\sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &,  \text{ else}\end{array}\right. </math>
+
|<math> \text{Taylor Series in one variable } = \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n}</math> [[Taylor_maclaurin_series|(info)]]
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | Infinite Geometric Series Formula || <math>\sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1}{1-x}&, \text{ if } |x|\leq 1\\ \text{diverges} &, \text{ else }\end{array}\right. </math>
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|<math> \text{Taylor Series in } d \text{ variables } =\sum_{n_1=0}^{\infin} \cdots \sum_{n_d=0}^{\infin}
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\frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + \cdots + n_d}f}{\partial x_1^{n_1}\cdots \partial x_d^{n_d}}\right)(a_1,\dots,a_d).\!</math>
 
|-
 
|-
−
! colspan="2" style="background: #eee;" | Other Series
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! colspan="2" style="background: #eee;" | [[Taylor_maclaurin_series|Taylor Series]] to remember
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | notes/name || equation
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|<math> \text{Exponential } e^x = \sum_{n=0}^\infty \frac{x^n}{n!}, \text{ for all } x\in {\mathbb C}\ </math>
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|-
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|<math> \text{Logarithm } \ln (1+x) = \sum^{\infin}_{n=1} (-1)^{n+1}\frac{x^n}n,\text{ when }-1<x\leq 1</math>
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|-
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|<math> \sin x \ = \ x \ - \ \frac{x^3}{3!} \ + \ \frac{x^5}{5!} \ - \ \frac{x^7}{7!} \ + \ \cdots, \quad \text{ for } - \infty < x < \infty</math>
 +
|-
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|<math> \cos x \ = \ 1  \ - \ \frac{x^2}{2!} \ + \ \frac{x^4}{4!} \ -  \ \frac{x6}{6!} \ + \ \cdots, \quad \text{ for } - \infty < x < \infty</math>
 +
|-
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! colspan="2" style="background: #eee;" | Geometric Series and related series
 +
|-
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|  [[more_on_geometric_series|(info)]] <math class="inline">  \text{Finite Geometric Series Formula } \sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &,  \text{ else}\end{array}\right. </math>
 +
|-
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|  [[more_on_geometric_series|(info)]] <math class="inline">  \text{Infinite Geometric Series Formula } \sum_{k=0}^\infty x^k = \left\{ \begin{array}{ll} \frac{1}{1-x}&, \text{ if } |x|\leq 1\\ \text{diverges} &, \text{ else }\end{array}\right. </math>
 +
|-
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|<math>\frac{x^m}{1-x} = \sum^{\infin}_{n=m} x^n, \quad\mbox{ for }|x| < 1 \text{ and } m\in\mathbb{N}_0\!</math>
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|-
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| <math>\frac{x}{(1-x)^2} = \sum^{\infin}_{n=1}n x^n, \quad\text{ for }|x| < 1\!</math>
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|-
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! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" |  Taylor series of Single Variable Functions
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|-
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| <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>
 +
|-
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|<math> \text{Rest of Lagrange } \qquad R_n = \frac {f^{(n)}(\zeta)(x-a)^n}{n!}</math>
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|-
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|<math> \text{Rest of Cauchy } \qquad R_n = \frac {f^{(n)}(\zeta)(x-\zeta)^{n-1}(x-a)}{(n-1)!}</math>
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|-
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! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" |  Binomial Series
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|-
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|For any positive integer n:
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|-
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| <math>
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\begin{align}
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(a+x)^n & = \sum_{k=0}^n \left( \begin{array}{ll}n\\k \end{array}\right) x^k a^{n-k}\\
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& = a^n + \binom{n}{1} a^{n-1}x + \binom{n}{2} a^{n-2}x^2 + \binom{n}{3} a^{n-3}x^3 + \ldots + x^n \\
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\end{align}
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</math>
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|-
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|For any complex number z:
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|-
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| <math>
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\begin{align}
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(a+x)^z & = a^z + za^{z-1}x + \frac {z(z-1)}{2!} a^{z-2}x^2 + \frac {z(z-1)(z-2)}{3!} a^{z-3}x^3 + \ldots \\
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& = a^z + \binom{z}{1} a^{z-1}x + \binom{z}{2} a^{z-2}x^2 + \binom{z}{3} a^{z-3}x^3 + \ldots \\
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\end{align}
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</math>
 +
|-
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| Some particular Cases:
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|-
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| <math> (a+x)^2 \ = \  a^2 \ + \  2ax \  + \  x^2</math>
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|-
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| <math> (a+x)^3 \ = \  a^3 \ + \  3a^2x \  + \  3ax^2 \ + \ x^3</math>
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|-
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| <math> (a+x)^4 \ = \  a^4 \ + \  4a^3x \  + \  6a^2x^2 \ + \ 4ax^3 \ + \ x^4</math>
 +
|-
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| <math> (1+x)^{-1} \ = \  1 \ - \  x \  + \  x^2 \ - \ x^3 \ + \ x^4 \ - \ \cdots  </math>
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|<math> -1 < x < 1 \qquad </math>
 +
|-
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| <math> (1+x)^{-2} \ = \  1 \ - \  2x \  + \  3x^2 \ - \ 4x^3 \ + \ 5x^4 \ - \ \cdots </math>
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|<math>  -1 < x < 1 \qquad</math>
 +
|-
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| <math> (1+x)^{-3} \ = \  1 \ - \  3x \  + \  6x^2 \ - \ 10x^3 \ + \ 15x^4 \ - \ \cdots  </math>
 +
|<math> -1 < x < 1 \qquad</math>
 +
|-
 +
| <math> (1+x)^{-1/2} \ = \  1 \ - \  \frac{1}{2}x \  + \  \frac{1 \cdot 3}{2 \cdot 4}x^2 \ - \ \frac {1 \cdot 3  \cdot 5 }{2 \cdot 4 \cdot 6} x^3 \ +  \ \cdots  </math>
 +
| <math> -1 < x  \leqq 1 \qquad </math>
 +
|-
 +
| <math> (1+x)^{1/2} \ = \  1 \ + \  \frac{1}{2}x \  - \  \frac{1 }{2 \cdot\ 4}x^2 \ + \ \frac {1 \cdot 3}{2 \cdot 4 \cdot 6} x^3 \ -  \ \cdots  </math>
 +
| <math> -1 < x  \leqq 1 \qquad </math>
 +
|-
 +
| <math> (1+x)^{-1/3} \ = \  1 \ - \  \frac{1}{3}x \  + \  \frac{1 \cdot 4}{3 \cdot 6}x^2 \ - \ \frac {1 \cdot 4  \cdot 7 }{3 \cdot 6 \cdot 9} x^3 \ +  \ \cdots  </math>
 +
| <math> -1 < x  \leqq 1 \qquad </math>
 +
|-
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| <math> (1+x)^{1/3} \ = \  1 \ + \  \frac{1}{3}x \  - \  \frac{2}{3 \cdot 6}x^2 \ + \ \frac {2 \cdot 5 }{3 \cdot 6 \cdot 9} x^3 \ -  \ \cdots  </math>
 +
| <math> -1 < x  \leqq 1 \qquad </math>
 +
|-
 +
! style="background: rgb(238, 238, 238) none ! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Series Expansion of Exponential functions and Logarithms
 +
|-
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| <math> e^x \ = \ 1  \ + \ x \ + \ \frac{x^2}{2!} \ + \ \frac{x^3}{3!} \ + \ \cdots</math>
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| <math> - \infty < x < \infty \qquad </math>
 +
|-
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| <math> a^x \ = \ e^{x \ln a} \ = \ 1  \ + \ x \ln a \ + \ \frac{(x \ln a)^2}{2!} \ + \ \frac{(x \ln a)^3}{3!} \ + \ \cdots</math>
 +
| <math> - \infty < x < \infty \qquad </math>
 +
|-
 +
| <math> \ln(1+x) \ = \ x \ - \ \frac{x^2}{2} \ + \ \frac{x^3}{3} \ - \ \frac{x^4}{4} \ + \ \cdots</math>
 +
| <math> -1 < x \leqq 1 \qquad </math>
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|-
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| <math> \frac{1}{2} \ln \left ( \frac {1+x}{1-x} \right ) \ = \ x \ + \ \frac{x^3}{3} \ + \ \frac {x^5}{5} \ + \ \frac{x^7}{7} \ + \ \cdots \ </math>
 +
|<math> -1 < x < 1 \qquad </math>
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|-
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|<math> \ln x \ = \ 2 \left \{ \left ( \frac {x-1}{x+1} \right ) \ + \ \frac{1}{3} \left ( \frac {x-1}{x+1} \right ) ^3 \ + \ \frac{1}{5} \left ( \frac{x-1}{x+1} \right ) ^ 5 \ + \ \cdots \ \right \} </math>
 +
|<math> x > 0 \qquad </math>
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|-
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|<math> \ln x \ = \ \left ( \frac {x-1}{x} \right ) \ + \ \frac{1}{2} \left ( \frac {x-1}{x} \right ) ^2 \ + \ \frac{1}{3} \left ( \frac{x-1}{x} \right ) ^ 3 \ + \ \cdots \  </math>
 +
|<math> x \geqq \frac {1}{2} \qquad </math>
 
|-
 
|-
−
|}
 
  
 +
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="3" |  Series Expansion of Circular functions
 +
|-
 +
| <math> \sin x \ = \ x \ - \ \frac{x^3}{3!} \ + \ \frac{x^5}{5!} \ - \ \frac{x^7}{7!} \ + \ \cdots</math>
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| <math> - \infty < x < \infty</math>
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|-
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| <math> \cos x \ = \ 1  \ - \ \frac{x^2}{2!} \ + \ \frac{x^4}{4!} \ -  \ \frac{x6}{6!} \ + \ \cdots</math>
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| <math> - \infty < x < \infty</math>
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|-
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| <math> \cot x \ = \ \frac{1}{x} \ - \ \frac {x}{3} \ - \ \frac{x^3}{45} \ - \ \frac{2x^5}{945} \ -  \  \cdots \ - \ \frac{2^{2n}B_n x^{2n-1}}{(2n)!} \ - \ \cdots </math>
 +
| <math> 0 < \left \vert x \right \vert < \pi \qquad </math>
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|-
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| <math> \frac{1}{\cos x} \ = \ 1 \ + \ \frac {x^2}{2} \ + \ \frac{x^4}{24} \ + \ \frac{61x^6}{720} \ +  \  \cdots \ - \ \frac{E_n x^{2n}}{(2n)!} \ + \ \cdots </math>
 +
| <math> \left \vert x \right \vert < \frac {\pi}{2} \qquad </math>
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|-
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| <math> \frac{1}{\sin x} \ = \ \frac{1}{x} \ + \ \frac {x}{6} \ + \ \frac{7x^3}{360} \ + \ \frac{31x^5}{15120} \ +  \  \cdots \ + \ \frac{2(2^{2n-1}-1)B_n x^{2n-1}}{(2n)!} \ + \ \cdots </math>
 +
| <math> 0 < \left \vert x \right \vert < \pi \qquad </math>
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|-
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|<math> \arcsin x = x + {1 \over 2}{x^3 \over 3} + \frac{1 \cdot 3}{ 2 \cdot 4} {x^5 \over 5} + \frac {1 \cdot 3 \cdot 5}{ 2 \cdot 4 \cdot 6}{x^7 \over 7} + \cdots </math>
 +
|<math> \left \vert x \right \vert < 1 \qquad</math>
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|-
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|<math> \arccos x = {\pi \over 2} - \sin ^{-1} x = {\pi \over 2} - \left ( x + {1 \over 2}{x^3 \over 3} +\frac{1 \cdot 3}{2 \cdot 4} {x^5 \over 5} + \cdots \ \right )</math>
 +
|<math>\left \vert x \right \vert < 1 \qquad</math>
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|-
 +
|<math> \arctan x =
 +
\begin{cases}
 +
x - {x^3 \over 3} + {x^5 \over 5} - { x^7 \over 7} + \cdots, & \left \vert x \right \vert < 1 \\
 +
{\pi \over 2} - {1 \over x} + {1 \over 3x^3} - {1 \over 5x^5} + \cdots, &\mbox{ if } x \geqq 1 \\
 +
-{\pi \over 2} - {1 \over x} + {1 \over 3x^3} - {1 \over 5x^5} + \cdots, &\mbox{ if } x \leqq -1
 +
\end{cases}
 +
</math>
 +
|-
 +
|<math> \arccot x = {\pi \over 2} - \arctan x =
 +
\begin{cases}
 +
{\pi \over 2} - \left ( x - {x^3 \over 3} + {x^5 \over 5} -  \cdots \right ), &\left \vert x \right \vert < 1 \\
 +
{\pi} + {1 \over x} - {1 \over 3x^3} + {1 \over 5x^5} - \cdots, & \mbox{ if } x > 1\\
 +
-{\pi} + {1 \over x} - {1 \over 3x^3} + {1 \over 5x^5} - \cdots, & \mbox{ if } x < -1
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\end{cases}
 +
</math>
 +
|-
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|<math> \arccos ({1 \over x}) = {\pi \over 2} - \left ( {1 \over x} + \frac{1}{2 \cdot 3 x^3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5 x^5} + \cdots \right )</math>
 +
|<math>\left \vert x \right \vert > 1 \qquad</math>
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|-
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|<math> \arcsin ({1 \over x}) = {1 \over x} + {1 \over 2 \cdot 3 x^3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5 x^5} + \cdots</math>
 +
|<math> \left \vert x \right \vert > 1</math>
 +
|-
 +
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" |  Series Expansion of Hyperbolic functions
 +
|-
 +
| <math>\, \sinh x  = x + {x^3 \over 3!} + {x^5 \over 5!} + { x^7 \over 7!} + \cdots\,</math>
 +
| <math> - \infty < x < \infty  \qquad</math>
 +
|-
 +
| <math>\, \cosh x  = 1 + {x^2 \over 2!} + {x^4 \over 4!} + { x^6 \over 6!} + \cdots\,</math>
 +
| <math> - \infty < x < \infty  \qquad</math>
 +
|-
 +
| <math>\, \tanh x  = x - {x^3 \over 3} + {2x^5 \over 15} - { 17x^7 \over 315} + \cdots  \ \frac{(-1)^{n-1}2^{2n}(2^{2n} -1)B_nx^{2n-1}}{(2n)!} + \cdots\,</math>
 +
| <math> \vert x \vert  < {\pi \over 2}  \qquad</math>
 +
|-
 +
| <math>\, \coth x  = {1 \over x} + {x \over 3} - {x^3 \over 45} + { 2x^5 \over 945} + \cdots \frac{(-1)^{n-1}2^{2n}b_nx^{2n-1}}{(2n)!} + \cdots\,</math>
 +
| <math> 0 < \vert x \vert < \pi  \qquad</math>
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|-
 +
|<math>\frac {1}{\cosh x} = 1 - {x2 \over 2} + {5x^4 \over 24} -{61x^6 \over 720} + \cdots \frac{(-1)^nE_nx^{2n}}{(2n)!} + \cdots</math>
 +
|<math>\vert x \vert < {\pi \over 2}</math>
 +
|-
 +
|<math> \frac{1}{\sinh x} = {1 \over x} - {x \over 6} + {7x^3 \over 360} - {31x^5 \over 15,120} + \cdots \frac{(-1)^n2(2^{2n-1}-1)B_nx^{2n-1}}{(2n)!} + \cdots</math>
 +
|<math>0 < \vert x \vert < \pi </math>
 +
|-
 +
|-
 +
|<math> \operatorname{arsinh}\,x =
 +
\begin{cases}
 +
  x - {x^3 \over 2 \cdot 3} + {1 \cdot 3 x^5 \cdot 2 \cdot 4 \cdot 5} -  {1 \cdot 3 \cdot 5 x^7 \over 2 \cdot 4 \cdot 6 \cdot 7} + \cdots,  & \left \vert x \right \vert < 1 \\
 +
\left ( \ln \vert 2x \vert + {1 \over 2 \cdot 2 x^2} - {1 \cdot 3 \over 2 \cdot 4 \cdot 4x^4} + {1 \cdot 3 \cdot 5 \over 2 \cdot 4 \cdot 6 \cdot 6x^6} - \cdots \right ), & x \geqq 1\\
 +
-\left ( \ln \vert 2x \vert + {1 \over 2 \cdot 2 x^2} - {1 \cdot 3 \over 2 \cdot 4 \cdot 4x^4} + {1 \cdot 3 \cdot 5 \over 2 \cdot 4 \cdot 6 \cdot 6x^6} - \cdots \right ), & x \leqq -1
 +
\end{cases}
 +
</math>
 +
|-
 +
|<math> \operatorname{arcosh} \,x = \begin{cases}
 +
  \{ \ln (2x) -  ( \frac{1}{2 \cdot 2x^2} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 4x^4} + \frac { 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 6x^6} + \cdots ) \}, & \operatorname{arsinh}\,x  > 0, x \geqq 1 \\
 +
- \{ \ln (2x) -  ( \frac{1}{2 \cdot 2x^2} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 4x^4} + \frac { 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 6x^6} + \cdots ) \}, & \operatorname{arsinh} \,x < 0, x \geqq 1
 +
\end{cases} </math>
 +
|-
 +
|<math> \operatorname{argth} \,x = x + { x^3 \over 5} + {x^5 \over 5 } + {x^7 \over 7 }+ \cdots </math>
 +
|<math> \vert x \vert < 1 \qquad</math>
 +
|-
 +
|<math> \operatorname{argcoth} \,x = {1 \over x} + { 1 \over 3x^3} + {1 \over 5x^5 } + {1 \over 7x^7 }+ \cdots </math>
 +
|<math> \vert x \vert > 1 \qquad</math>
 +
|-
 +
 +
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" |  Various Series
 +
|-
 +
| <math>\, e^{\sin x} = 1 + x + {x^2 \over 2} - {x^4 \over 8} - {x^5 \over 15} + \cdots\, </math>
 +
|<math> - \infty < x < \infty </math>
 +
|-
 +
| <math>\, e^{\cos x} = e \left ( 1 - {x^2 \over 2} + {x^4 \over 6} - {31x^6 \over 720} + \cdots \right ) \, </math>
 +
|<math> - \infty < x < \infty </math>
 +
|-
 +
| <math>\, e^{\tan x} = 1 + x + {x^2 \over 2} + {x^3 \over 2} + {3x^4 \over 8} + \cdots  \, </math>
 +
|<math> \vert x \vert < { \pi \over 2} </math>
 +
|-
 +
|<math> e^x \sin x = x + x^2 + {2x^3 \over 3 } - {x^5 \over 30} - {x^6 \over 90} + \cdots + \frac{2^{n/2} \sin (n \pi /4)\ x^n}{n!} + \cdots</math>
 +
|<math> - \infty < x < \infty</math>
 +
|-
 +
|<math> e^x \cos x = 1 + x - {x^3 \over 3 } - {x^4 \over 6} + \cdots + \frac{2^{n/2} \cos (n \pi /4)\ x^n}{n!} + \cdots</math>
 +
|<math> - \infty < x < \infty</math>
 +
|-
 +
|<math>\ln \vert \sin x \vert = \ln \vert x \vert - {x^2 \over 6} - {x^4 \over 180} - {x^6 \over 2835} - \cdots - \frac{2^{2n-1}B_nx^{2n}}{n(2n)!} + \cdots </math>
 +
|<math> 0 < \vert x \vert < \pi</math>
 +
|-
 +
|<math>\ln \vert \cos x \vert =  - {x^2 \over 2} - {x^4 \over 12} - {x^6 \over 45} - {17x^8 \over 2520} - \cdots - \frac{2^{2n-1}(2^{2n}-1)B_nx^{2n}}{n(2n)!} + \cdots </math>
 +
|<math> \vert x \vert < {\pi \over 2}</math>
 +
|-
 +
|<math>\ln \vert \tan x \vert =  \ln \vert x \vert + {x^2 \over 3} + {7x^4 \over 90} + {62x^6 \over 2835}+ \cdots + \frac{2^{2n}(2^{2n-1}-1)B_nx^{2n}}{n(2n)!} + \cdots </math>
 +
|<math> 0 < \vert x \vert < {\pi \over 2}</math>
 +
|-
 +
|<math> \frac{\ln (1+x)}{1+x} = x - (1+ {1 \over 2})^{x^2} + (1 + {1 \over 2} + {1 \over 3})^{x^3} - \cdots </math>
 +
|<math>\vert x \vert < 1</math>
 +
|
 +
|-
 +
! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" |  Series of Reciprocal Power Series
 +
|-
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| <math>\text{if }\ y = c_1x +c_2x^3 +c_3x^3 + c_4x^4 + c_5x^5 + c_6x^6 + \cdots\,\qquad \text{then }\ x = C_1y+C_2y^2+C_3y^3+C_4y^4+C_5y^5+C_6y^6+\cdots </math>
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|<math>\text{where }\ c_1C_1 = 1, \qquad c_1^3C_2= -c_2, \qquad c_1^7C_3 = 2c_2^2 - c_1c_3 </math>
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|<math>c_1^7C_4 = 5c_1c_2c_3 - 5c_2^3 - c_2^2c_4, \qquad c_1^9C_5 = 6c_1^2c_2c_4 + </math>
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|<math>c_1^{11}C_6 = 7 c_1^3c_2 c_5 + 84 c_1 c_2^3c_3 + 7c_1^3c_3c_4 - 28c_1^2c_2c_3^2 - c_1^4c/-6 - 28c_1^2c_2^2c_4 - 42c_2^5 </math>
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! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" |  Taylor Series of Two Variables function
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| <math>\, f(x,y) = f(a,b) + (x-a)f_x(a,b) + (y-b)f_y(a,b) + </math>
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|align ="center" |<math>  {1 \over 2!} \left \{ (x-a)^2f_{xx}(a,b) + 2(x-a)(y-b)f_{xy}(a,b)+(y-b)^2f_{yy}(a,b) \right \} + \cdots\,</math>
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|<math> f_x(a,b),f_y(a,b) , \cdots \text {denote the partial derivatives with respect to } x ,\ y \cdots  </math>
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[[Collective_Table_of_Formulas|Back to Collective Table]]

Latest revision as of 17:14, 27 February 2015


keywords: Taylor, Geometric, Binomial

Collective Table of Formulas

Power Series

(Used in ECE301 and ECE438)

Taylor Series Formulas
Series in symbolic forms
$ \text{Taylor Series in one variable } = \sum_{n=0} ^ {\infin } \frac {f^{(n)}(a)}{n!} \, (x-a)^{n} $ (info)
$ \text{Taylor Series in } d \text{ variables } =\sum_{n_1=0}^{\infin} \cdots \sum_{n_d=0}^{\infin} \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + \cdots + n_d}f}{\partial x_1^{n_1}\cdots \partial x_d^{n_d}}\right)(a_1,\dots,a_d).\! $
Taylor Series to remember
$ \text{Exponential } e^x = \sum_{n=0}^\infty \frac{x^n}{n!}, \text{ for all } x\in {\mathbb C}\ $
$ \text{Logarithm } \ln (1+x) = \sum^{\infin}_{n=1} (-1)^{n+1}\frac{x^n}n,\text{ when }-1<x\leq 1 $
$ \sin x \ = \ x \ - \ \frac{x^3}{3!} \ + \ \frac{x^5}{5!} \ - \ \frac{x^7}{7!} \ + \ \cdots, \quad \text{ for } - \infty < x < \infty $
$ \cos x \ = \ 1 \ - \ \frac{x^2}{2!} \ + \ \frac{x^4}{4!} \ - \ \frac{x6}{6!} \ + \ \cdots, \quad \text{ for } - \infty < x < \infty $
Geometric Series and related series
(info) $ \text{Finite Geometric Series Formula } \sum_{k=0}^n x^k = \left\{ \begin{array}{ll} \frac{1-x^{n+1}}{1-x}&, \text{ if } x\neq 1\\ n+1 &, \text{ else}\end{array}\right. $
(info) $ \text{Infinite Geometric Series Formula } \sum_{k=0}^\infty x^k = \left\{ \begin{array}{ll} \frac{1}{1-x}&, \text{ if } |x|\leq 1\\ \text{diverges} &, \text{ else }\end{array}\right. $
$ \frac{x^m}{1-x} = \sum^{\infin}_{n=m} x^n, \quad\mbox{ for }|x| < 1 \text{ and } m\in\mathbb{N}_0\! $
$ \frac{x}{(1-x)^2} = \sum^{\infin}_{n=1}n x^n, \quad\text{ for }|x| < 1\! $
Taylor series of Single Variable Functions
$ \,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 \, $
$ \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)!} $
Binomial Series
For any positive integer n:
$ \begin{align} (a+x)^n & = \sum_{k=0}^n \left( \begin{array}{ll}n\\k \end{array}\right) x^k a^{n-k}\\ & = a^n + \binom{n}{1} a^{n-1}x + \binom{n}{2} a^{n-2}x^2 + \binom{n}{3} a^{n-3}x^3 + \ldots + x^n \\ \end{align} $
For any complex number z:
$ \begin{align} (a+x)^z & = a^z + za^{z-1}x + \frac {z(z-1)}{2!} a^{z-2}x^2 + \frac {z(z-1)(z-2)}{3!} a^{z-3}x^3 + \ldots \\ & = a^z + \binom{z}{1} a^{z-1}x + \binom{z}{2} a^{z-2}x^2 + \binom{z}{3} a^{z-3}x^3 + \ldots \\ \end{align} $
Some particular Cases:
$ (a+x)^2 \ = \ a^2 \ + \ 2ax \ + \ x^2 $
$ (a+x)^3 \ = \ a^3 \ + \ 3a^2x \ + \ 3ax^2 \ + \ x^3 $
$ (a+x)^4 \ = \ a^4 \ + \ 4a^3x \ + \ 6a^2x^2 \ + \ 4ax^3 \ + \ x^4 $
$ (1+x)^{-1} \ = \ 1 \ - \ x \ + \ x^2 \ - \ x^3 \ + \ x^4 \ - \ \cdots $ $ -1 < x < 1 \qquad $
$ (1+x)^{-2} \ = \ 1 \ - \ 2x \ + \ 3x^2 \ - \ 4x^3 \ + \ 5x^4 \ - \ \cdots $ $ -1 < x < 1 \qquad $
$ (1+x)^{-3} \ = \ 1 \ - \ 3x \ + \ 6x^2 \ - \ 10x^3 \ + \ 15x^4 \ - \ \cdots $ $ -1 < x < 1 \qquad $
$ (1+x)^{-1/2} \ = \ 1 \ - \ \frac{1}{2}x \ + \ \frac{1 \cdot 3}{2 \cdot 4}x^2 \ - \ \frac {1 \cdot 3 \cdot 5 }{2 \cdot 4 \cdot 6} x^3 \ + \ \cdots $ $ -1 < x \leqq 1 \qquad $
$ (1+x)^{1/2} \ = \ 1 \ + \ \frac{1}{2}x \ - \ \frac{1 }{2 \cdot\ 4}x^2 \ + \ \frac {1 \cdot 3}{2 \cdot 4 \cdot 6} x^3 \ - \ \cdots $ $ -1 < x \leqq 1 \qquad $
$ (1+x)^{-1/3} \ = \ 1 \ - \ \frac{1}{3}x \ + \ \frac{1 \cdot 4}{3 \cdot 6}x^2 \ - \ \frac {1 \cdot 4 \cdot 7 }{3 \cdot 6 \cdot 9} x^3 \ + \ \cdots $ $ -1 < x \leqq 1 \qquad $
$ (1+x)^{1/3} \ = \ 1 \ + \ \frac{1}{3}x \ - \ \frac{2}{3 \cdot 6}x^2 \ + \ \frac {2 \cdot 5 }{3 \cdot 6 \cdot 9} x^3 \ - \ \cdots $ $ -1 < x \leqq 1 \qquad $
Series Expansion of Exponential functions and Logarithms
$ e^x \ = \ 1 \ + \ x \ + \ \frac{x^2}{2!} \ + \ \frac{x^3}{3!} \ + \ \cdots $ $ - \infty < x < \infty \qquad $
$ a^x \ = \ e^{x \ln a} \ = \ 1 \ + \ x \ln a \ + \ \frac{(x \ln a)^2}{2!} \ + \ \frac{(x \ln a)^3}{3!} \ + \ \cdots $ $ - \infty < x < \infty \qquad $
$ \ln(1+x) \ = \ x \ - \ \frac{x^2}{2} \ + \ \frac{x^3}{3} \ - \ \frac{x^4}{4} \ + \ \cdots $ $ -1 < x \leqq 1 \qquad $
$ \frac{1}{2} \ln \left ( \frac {1+x}{1-x} \right ) \ = \ x \ + \ \frac{x^3}{3} \ + \ \frac {x^5}{5} \ + \ \frac{x^7}{7} \ + \ \cdots \ $ $ -1 < x < 1 \qquad $
$ \ln x \ = \ 2 \left \{ \left ( \frac {x-1}{x+1} \right ) \ + \ \frac{1}{3} \left ( \frac {x-1}{x+1} \right ) ^3 \ + \ \frac{1}{5} \left ( \frac{x-1}{x+1} \right ) ^ 5 \ + \ \cdots \ \right \} $ $ x > 0 \qquad $
$ \ln x \ = \ \left ( \frac {x-1}{x} \right ) \ + \ \frac{1}{2} \left ( \frac {x-1}{x} \right ) ^2 \ + \ \frac{1}{3} \left ( \frac{x-1}{x} \right ) ^ 3 \ + \ \cdots \ $ $ x \geqq \frac {1}{2} \qquad $
Series Expansion of Circular functions
$ \sin x \ = \ x \ - \ \frac{x^3}{3!} \ + \ \frac{x^5}{5!} \ - \ \frac{x^7}{7!} \ + \ \cdots $ $ - \infty < x < \infty $
$ \cos x \ = \ 1 \ - \ \frac{x^2}{2!} \ + \ \frac{x^4}{4!} \ - \ \frac{x6}{6!} \ + \ \cdots $ $ - \infty < x < \infty $
$ \cot x \ = \ \frac{1}{x} \ - \ \frac {x}{3} \ - \ \frac{x^3}{45} \ - \ \frac{2x^5}{945} \ - \ \cdots \ - \ \frac{2^{2n}B_n x^{2n-1}}{(2n)!} \ - \ \cdots $ $ 0 < \left \vert x \right \vert < \pi \qquad $
$ \frac{1}{\cos x} \ = \ 1 \ + \ \frac {x^2}{2} \ + \ \frac{x^4}{24} \ + \ \frac{61x^6}{720} \ + \ \cdots \ - \ \frac{E_n x^{2n}}{(2n)!} \ + \ \cdots $ $ \left \vert x \right \vert < \frac {\pi}{2} \qquad $
$ \frac{1}{\sin x} \ = \ \frac{1}{x} \ + \ \frac {x}{6} \ + \ \frac{7x^3}{360} \ + \ \frac{31x^5}{15120} \ + \ \cdots \ + \ \frac{2(2^{2n-1}-1)B_n x^{2n-1}}{(2n)!} \ + \ \cdots $ $ 0 < \left \vert x \right \vert < \pi \qquad $
$ \arcsin x = x + {1 \over 2}{x^3 \over 3} + \frac{1 \cdot 3}{ 2 \cdot 4} {x^5 \over 5} + \frac {1 \cdot 3 \cdot 5}{ 2 \cdot 4 \cdot 6}{x^7 \over 7} + \cdots $ $ \left \vert x \right \vert < 1 \qquad $
$ \arccos x = {\pi \over 2} - \sin ^{-1} x = {\pi \over 2} - \left ( x + {1 \over 2}{x^3 \over 3} +\frac{1 \cdot 3}{2 \cdot 4} {x^5 \over 5} + \cdots \ \right ) $ $ \left \vert x \right \vert < 1 \qquad $
$ \arctan x = \begin{cases} x - {x^3 \over 3} + {x^5 \over 5} - { x^7 \over 7} + \cdots, & \left \vert x \right \vert < 1 \\ {\pi \over 2} - {1 \over x} + {1 \over 3x^3} - {1 \over 5x^5} + \cdots, &\mbox{ if } x \geqq 1 \\ -{\pi \over 2} - {1 \over x} + {1 \over 3x^3} - {1 \over 5x^5} + \cdots, &\mbox{ if } x \leqq -1 \end{cases} $
$ \arccot x = {\pi \over 2} - \arctan x = \begin{cases} {\pi \over 2} - \left ( x - {x^3 \over 3} + {x^5 \over 5} - \cdots \right ), &\left \vert x \right \vert < 1 \\ {\pi} + {1 \over x} - {1 \over 3x^3} + {1 \over 5x^5} - \cdots, & \mbox{ if } x > 1\\ -{\pi} + {1 \over x} - {1 \over 3x^3} + {1 \over 5x^5} - \cdots, & \mbox{ if } x < -1 \end{cases} $
$ \arccos ({1 \over x}) = {\pi \over 2} - \left ( {1 \over x} + \frac{1}{2 \cdot 3 x^3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5 x^5} + \cdots \right ) $ $ \left \vert x \right \vert > 1 \qquad $
$ \arcsin ({1 \over x}) = {1 \over x} + {1 \over 2 \cdot 3 x^3} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 5 x^5} + \cdots $ $ \left \vert x \right \vert > 1 $
Series Expansion of Hyperbolic functions
$ \, \sinh x = x + {x^3 \over 3!} + {x^5 \over 5!} + { x^7 \over 7!} + \cdots\, $ $ - \infty < x < \infty \qquad $
$ \, \cosh x = 1 + {x^2 \over 2!} + {x^4 \over 4!} + { x^6 \over 6!} + \cdots\, $ $ - \infty < x < \infty \qquad $
$ \, \tanh x = x - {x^3 \over 3} + {2x^5 \over 15} - { 17x^7 \over 315} + \cdots \ \frac{(-1)^{n-1}2^{2n}(2^{2n} -1)B_nx^{2n-1}}{(2n)!} + \cdots\, $ $ \vert x \vert < {\pi \over 2} \qquad $
$ \, \coth x = {1 \over x} + {x \over 3} - {x^3 \over 45} + { 2x^5 \over 945} + \cdots \frac{(-1)^{n-1}2^{2n}b_nx^{2n-1}}{(2n)!} + \cdots\, $ $ 0 < \vert x \vert < \pi \qquad $
$ \frac {1}{\cosh x} = 1 - {x2 \over 2} + {5x^4 \over 24} -{61x^6 \over 720} + \cdots \frac{(-1)^nE_nx^{2n}}{(2n)!} + \cdots $ $ \vert x \vert < {\pi \over 2} $
$ \frac{1}{\sinh x} = {1 \over x} - {x \over 6} + {7x^3 \over 360} - {31x^5 \over 15,120} + \cdots \frac{(-1)^n2(2^{2n-1}-1)B_nx^{2n-1}}{(2n)!} + \cdots $ $ 0 < \vert x \vert < \pi $
$ \operatorname{arsinh}\,x = \begin{cases} x - {x^3 \over 2 \cdot 3} + {1 \cdot 3 x^5 \cdot 2 \cdot 4 \cdot 5} - {1 \cdot 3 \cdot 5 x^7 \over 2 \cdot 4 \cdot 6 \cdot 7} + \cdots, & \left \vert x \right \vert < 1 \\ \left ( \ln \vert 2x \vert + {1 \over 2 \cdot 2 x^2} - {1 \cdot 3 \over 2 \cdot 4 \cdot 4x^4} + {1 \cdot 3 \cdot 5 \over 2 \cdot 4 \cdot 6 \cdot 6x^6} - \cdots \right ), & x \geqq 1\\ -\left ( \ln \vert 2x \vert + {1 \over 2 \cdot 2 x^2} - {1 \cdot 3 \over 2 \cdot 4 \cdot 4x^4} + {1 \cdot 3 \cdot 5 \over 2 \cdot 4 \cdot 6 \cdot 6x^6} - \cdots \right ), & x \leqq -1 \end{cases} $
$ \operatorname{arcosh} \,x = \begin{cases} \{ \ln (2x) - ( \frac{1}{2 \cdot 2x^2} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 4x^4} + \frac { 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 6x^6} + \cdots ) \}, & \operatorname{arsinh}\,x > 0, x \geqq 1 \\ - \{ \ln (2x) - ( \frac{1}{2 \cdot 2x^2} + \frac{1 \cdot 3}{2 \cdot 4 \cdot 4x^4} + \frac { 1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6 \cdot 6x^6} + \cdots ) \}, & \operatorname{arsinh} \,x < 0, x \geqq 1 \end{cases} $
$ \operatorname{argth} \,x = x + { x^3 \over 5} + {x^5 \over 5 } + {x^7 \over 7 }+ \cdots $ $ \vert x \vert < 1 \qquad $
$ \operatorname{argcoth} \,x = {1 \over x} + { 1 \over 3x^3} + {1 \over 5x^5 } + {1 \over 7x^7 }+ \cdots $ $ \vert x \vert > 1 \qquad $
Various Series
$ \, e^{\sin x} = 1 + x + {x^2 \over 2} - {x^4 \over 8} - {x^5 \over 15} + \cdots\, $ $ - \infty < x < \infty $
$ \, e^{\cos x} = e \left ( 1 - {x^2 \over 2} + {x^4 \over 6} - {31x^6 \over 720} + \cdots \right ) \, $ $ - \infty < x < \infty $
$ \, e^{\tan x} = 1 + x + {x^2 \over 2} + {x^3 \over 2} + {3x^4 \over 8} + \cdots \, $ $ \vert x \vert < { \pi \over 2} $
$ e^x \sin x = x + x^2 + {2x^3 \over 3 } - {x^5 \over 30} - {x^6 \over 90} + \cdots + \frac{2^{n/2} \sin (n \pi /4)\ x^n}{n!} + \cdots $ $ - \infty < x < \infty $
$ e^x \cos x = 1 + x - {x^3 \over 3 } - {x^4 \over 6} + \cdots + \frac{2^{n/2} \cos (n \pi /4)\ x^n}{n!} + \cdots $ $ - \infty < x < \infty $
$ \ln \vert \sin x \vert = \ln \vert x \vert - {x^2 \over 6} - {x^4 \over 180} - {x^6 \over 2835} - \cdots - \frac{2^{2n-1}B_nx^{2n}}{n(2n)!} + \cdots $ $ 0 < \vert x \vert < \pi $
$ \ln \vert \cos x \vert = - {x^2 \over 2} - {x^4 \over 12} - {x^6 \over 45} - {17x^8 \over 2520} - \cdots - \frac{2^{2n-1}(2^{2n}-1)B_nx^{2n}}{n(2n)!} + \cdots $ $ \vert x \vert < {\pi \over 2} $
$ \ln \vert \tan x \vert = \ln \vert x \vert + {x^2 \over 3} + {7x^4 \over 90} + {62x^6 \over 2835}+ \cdots + \frac{2^{2n}(2^{2n-1}-1)B_nx^{2n}}{n(2n)!} + \cdots $ $ 0 < \vert x \vert < {\pi \over 2} $
$ \frac{\ln (1+x)}{1+x} = x - (1+ {1 \over 2})^{x^2} + (1 + {1 \over 2} + {1 \over 3})^{x^3} - \cdots $ $ \vert x \vert < 1 $
Series of Reciprocal Power Series
$ \text{if }\ y = c_1x +c_2x^3 +c_3x^3 + c_4x^4 + c_5x^5 + c_6x^6 + \cdots\,\qquad \text{then }\ x = C_1y+C_2y^2+C_3y^3+C_4y^4+C_5y^5+C_6y^6+\cdots $
$ \text{where }\ c_1C_1 = 1, \qquad c_1^3C_2= -c_2, \qquad c_1^7C_3 = 2c_2^2 - c_1c_3 $
$ c_1^7C_4 = 5c_1c_2c_3 - 5c_2^3 - c_2^2c_4, \qquad c_1^9C_5 = 6c_1^2c_2c_4 + $
$ c_1^{11}C_6 = 7 c_1^3c_2 c_5 + 84 c_1 c_2^3c_3 + 7c_1^3c_3c_4 - 28c_1^2c_2c_3^2 - c_1^4c/-6 - 28c_1^2c_2^2c_4 - 42c_2^5 $
Taylor Series of Two Variables function
$ \, f(x,y) = f(a,b) + (x-a)f_x(a,b) + (y-b)f_y(a,b) + $
$ {1 \over 2!} \left \{ (x-a)^2f_{xx}(a,b) + 2(x-a)(y-b)f_{xy}(a,b)+(y-b)^2f_{yy}(a,b) \right \} + \cdots\, $
$ f_x(a,b),f_y(a,b) , \cdots \text {denote the partial derivatives with respect to } x ,\ y \cdots $

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Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva