Chip 1998 January/CHIPCD1_98.iso/software/nowosci/edwzorow/wzory.wiz

Text File | 1997-09-08 | 23KB | 2,745 lines

@EQEDWIZ@
Ca│ka oznaczona
\EQEDint
{\EQEDplain
{a}
}
{\EQEDplain
{b}
}
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\EQEDplain
{dx}
}
@
wsp≤│czynniki Fouriera (wzory Eulera)
\EQEDlo
{\EQEDplain
{k}
}
{\EQEDplain
{a}
}
\EQEDplain
{=}
\frac
{\EQEDplain
{2}
}
{\EQEDplain
{T}
}
\EQEDint
{\EQEDplain
{0}
}
{\EQEDplain
{T}
}
{\EQEDplain
{f}
}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\cos \EQEDplain
{k}
\omega \EQEDplain
{xdx}

\EQEDlo
{\EQEDplain
{k}
}
{\EQEDplain
{b}
}
\EQEDplain
{=}
\frac
{\EQEDplain
{2}
}
{\EQEDplain
{T}
}
\EQEDint
{\EQEDplain
{0}
}
{\EQEDplain
{T}
}
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\sin \EQEDplain
{k}
\omega \EQEDplain
{xdx}
}
@
zastapienie funkcji szeregami Fouriera
\EQEDlo
{\EQEDplain
{n}
}
{\EQEDplain
{s}
}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\EQEDplain
{=}
\frac
{\EQEDplain
{1}
}
{\EQEDplain
{2}
}
\EQEDlo
{\EQEDplain
{0}
}
{\EQEDplain
{a}
}
\EQEDplain
{+}
\EQEDsum
{\EQEDplain
{k=1}
}
{\EQEDplain
{n}
}
{\EQEDlo
{\EQEDplain
{k}
}
{\EQEDplain
{a}
}
\cos \EQEDplain
{k}
\omega \EQEDplain
{x}
\EQEDplain
{+}
\EQEDsum
{\EQEDplain
{k=1}
}
{\EQEDplain
{n}
}
{\EQEDlo
{\EQEDplain
{k}
}
{\EQEDplain
{b}
}
\sin \EQEDplain
{k}
\omega \EQEDplain
{x}
\EQEDplain
{}
}
}
@
szereg Taylora
\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\EQEDplain
{=}
\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDplain
{a}
}
{\right)}
\EQEDplain
{+}
\frac
{\EQEDplain
{x-a}
}
{\EQEDplain
{1!}
}
\EQEDhi
{\EQEDplain
{,}
}
{\EQEDplain
{f}
}
\EQEDbrackets
{\left(}
{\EQEDplain
{a}
}
{\right)}
\EQEDplain
{+}
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDbrackets
{\left(}
{\EQEDplain
{x-a}
}
{\right)}
}
}
{\EQEDplain
{2!}
}
\EQEDhi
{\EQEDplain
{,,}
}
{\EQEDplain
{f}
}
\EQEDbrackets
{\left(}
{\EQEDplain
{a}
}
{\right)}
\EQEDplain
{+}
\EQEDplain
{...}
\EQEDplain
{+}
\frac
{\EQEDhi
{\EQEDplain
{n}
}
{\EQEDbrackets
{\left(}
{\EQEDplain
{x-a}
}
{\right)}
}
}
{\EQEDplain
{n!}
}
\EQEDhi
{\EQEDplain
{n}
}
{\EQEDplain
{f}
}
\EQEDbrackets
{\left(}
{\EQEDplain
{a}
}
{\right)}
@
szereg Maclaurina
\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\EQEDplain
{=}
\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDplain
{0}
}
{\right)}
\EQEDplain
{+}
\frac
{\EQEDplain
{x}
}
{\EQEDplain
{1!}
}
\EQEDhi
{\EQEDplain
{,}
}
{\EQEDplain
{f}
}
\EQEDbrackets
{\left(}
{\EQEDplain
{0}
}
{\right)}
\EQEDplain
{+}
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{x}
}
}
{\EQEDplain
{2!}
}
\EQEDhi
{\EQEDplain
{,,}
}
{\EQEDplain
{f}
}
\EQEDbrackets
{\left(}
{\EQEDplain
{0}
}
{\right)}
\EQEDplain
{+}
\EQEDplain
{...}
\EQEDplain
{+}
\frac
{\EQEDhi
{\EQEDplain
{n}
}
{\EQEDplain
{x}
}
}
{\EQEDplain
{n!}
}
\EQEDhi
{\EQEDplain
{n}
}
{\EQEDplain
{f}
}
\EQEDbrackets
{\left(}
{\EQEDplain
{0}
}
{\right)}
\EQEDplain
{+}
\EQEDplain
{...}
@
wz≤r Newtona
\EQEDhi
{\EQEDplain
{n}
}
{\EQEDbrackets
{\left(}
{\EQEDplain
{a+b}
}
{\right)}
}
\EQEDplain
{=}
\EQEDhi
{\EQEDplain
{n}
}
{\EQEDplain
{a}
}
\EQEDplain
{+}
\EQEDhi
{\EQEDplain
{n-1}
}
{\EQEDplain
{na}
}
\EQEDplain
{b}
\EQEDplain
{+}
\frac
{\EQEDplain
{n}
\EQEDbrackets
{\left(}
{\EQEDplain
{n-1}
}
{\right)}
}
{\EQEDplain
{2!}
}
\EQEDhi
{\EQEDplain
{n-2}
}
{\EQEDplain
{a}
}
\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{b}
}
\EQEDplain
{+}
\frac
{\EQEDplain
{n}
\EQEDbrackets
{\left(}
{\EQEDplain
{n-1}
}
{\right)}
\EQEDbrackets
{\left(}
{\EQEDplain
{n-2}
}
{\right)}
}
{\EQEDplain
{3!}
}
\EQEDhi
{\EQEDplain
{n-3}
}
{\EQEDplain
{a}
}
\EQEDhi
{\EQEDplain
{3}
}
{\EQEDplain
{b}
}
\EQEDplain
{+...+}
\frac
{\EQEDplain
{n}
\EQEDbrackets
{\left(}
{\EQEDplain
{n-1}
}
{\right)}
\EQEDplain
{...}
\EQEDbrackets
{\left(}
{\EQEDplain
{n-m+1}
}
{\right)}
}
{\EQEDplain
{3!}
}
\EQEDhi
{\EQEDplain
{n-m}
}
{\EQEDplain
{a}
}
\EQEDhi
{\EQEDplain
{m}
}
{\EQEDplain
{b}
}
\EQEDplain
{+...+}
\EQEDhi
{\EQEDplain
{n}
}
{\EQEDplain
{b}
}
@
nier≤wno£µ Cauchy'ego
\frac
{\EQEDlo
{\EQEDplain
{1}
}
{\EQEDplain
{a}
}
\EQEDplain
{+}
\EQEDlo
{\EQEDplain
{2}
}
{\EQEDplain
{a}
}
\EQEDplain
{+}
\EQEDplain
{...}
\EQEDplain
{+}
\EQEDlo
{\EQEDplain
{n}
}
{\EQEDplain
{a}
}
}
{\EQEDplain
{n}
}
\geq \EQEDroot
{\EQEDplain
{n}
}
{\EQEDlo
{\EQEDplain
{1}
}
{\EQEDplain
{a}
}
\EQEDlo
{\EQEDplain
{2}
}
{\EQEDplain
{a}
}
\EQEDplain
{...}
\EQEDlo
{\EQEDplain
{n}
}
{\EQEDplain
{a}
}
}
@
nier≤wno£µ Buniakowskiego-Cauchy'ego
\EQEDlo
{\EQEDplain
{1}
}
{\EQEDplain
{a}
}
\EQEDlo
{\EQEDplain
{1}
}
{\EQEDplain
{b}
}
\EQEDplain
{+}
\EQEDlo
{\EQEDplain
{2}
}
{\EQEDplain
{a}
}
\EQEDlo
{\EQEDplain
{2}
}
{\EQEDplain
{b}
}
\EQEDplain
{+}
\EQEDlo
{\EQEDplain
{3}
}
{\EQEDplain
{a}
}
\EQEDlo
{\EQEDplain
{3}
}
{\EQEDplain
{b}
}
\EQEDplain
{+}
\EQEDplain
{...}
\EQEDplain
{+}
\EQEDlo
{\EQEDplain
{n}
}
{\EQEDplain
{a}
}
\EQEDlo
{\EQEDplain
{n}
}
{\EQEDplain
{b}
}
\leq \EQEDroot
{\EQEDplain
{}
}
{\EQEDboth
{\EQEDplain
{1}
}
{\EQEDplain
{2}
}
{\EQEDplain
{a}
}
\EQEDplain
{+}
\EQEDboth
{\EQEDplain
{2}
}
{\EQEDplain
{2}
}
{\EQEDplain
{a}
}
\EQEDplain
{+...+}
\EQEDboth
{\EQEDplain
{n}
}
{\EQEDplain
{2}
}
{\EQEDplain
{a}
}
}
\EQEDroot
{\EQEDplain
{}
}
{\EQEDboth
{\EQEDplain
{1}
}
{\EQEDplain
{2}
}
{\EQEDplain
{b}
}
\EQEDplain
{+}
\EQEDboth
{\EQEDplain
{2}
}
{\EQEDplain
{2}
}
{\EQEDplain
{b}
}
\EQEDplain
{+...+}
\EQEDboth
{\EQEDplain
{n}
}
{\EQEDplain
{2}
}
{\EQEDplain
{b}
}
}
@
wyraz og≤lny ci╣gu arytmetycznego
\EQEDlo
{\EQEDplain
{n}
}
{\EQEDplain
{a}
}
\EQEDplain
{=}
\EQEDlo
{\EQEDplain
{1}
}
{\EQEDplain
{a}
}
\EQEDplain
{+}
\EQEDbrackets
{\left(}
{\EQEDplain
{n-1}
}
{\right)}
\EQEDplain
{r}
@
suma n wyraz≤w ci╣gu arytmetycznego
\EQEDlo
{\EQEDplain
{n}
}
{\EQEDplain
{S}
}
\EQEDplain
{=}
\frac
{\EQEDplain
{n}
\EQEDbrackets
{\left(}
{\EQEDlo
{\EQEDplain
{1}
}
{\EQEDplain
{a}
}
\EQEDplain
{+}
\EQEDlo
{\EQEDplain
{n}
}
{\EQEDplain
{a}
}
}
{\right)}
}
{\EQEDplain
{2}
}
@
wyraz og≤lny ci╣gu geometrycznego
\EQEDlo
{\EQEDplain
{n}
}
{\EQEDplain
{a}
}
\EQEDplain
{=}
\EQEDlo
{\EQEDplain
{1}
}
{\EQEDplain
{a}
}
\EQEDhi
{\EQEDplain
{n-1}
}
{\EQEDplain
{q}
}
@
suma szeregu geometrycznego
\EQEDlo
{\EQEDplain
{n}
}
{\EQEDplain
{S}
}
\EQEDplain
{=}
\frac
{\EQEDlo
{\EQEDplain
{1}
}
{\EQEDplain
{a}
}
\EQEDbrackets
{\left(}
{\EQEDhi
{\EQEDplain
{n}
}
{\EQEDplain
{q}
}
\EQEDplain
{-}
\EQEDplain
{1}
}
{\right)}
}
{\EQEDplain
{q-1}
}
\EQEDplain
{,}
\EQEDplain
{q}
\neq \EQEDplain
{1}

\EQEDlo
{\EQEDplain
{n}
}
{\EQEDplain
{S}
}
\EQEDplain
{=}
\EQEDplain
{n}
\EQEDlo
{\EQEDplain
{1}
}
{\EQEDplain
{a}
}
\EQEDplain
{,}
\EQEDplain
{q}
\EQEDplain
{=}
\EQEDplain
{1}
@
funkcja gamma
\Gamma \EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\EQEDplain
{=}
\EQEDlim
{\lim}
{\EQEDplain
{n}
\rightarrow \infty }
{\frac
{\EQEDplain
{n!}
\EQEDhi
{\EQEDplain
{x-1}
}
{\EQEDplain
{n}
}
}
{\EQEDplain
{x}
\EQEDbrackets
{\left(}
{\EQEDplain
{x+1}
}
{\right)}
\EQEDbrackets
{\left(}
{\EQEDplain
{x+2}
}
{\right)}
\EQEDplain
{...}
\EQEDbrackets
{\left(}
{\EQEDplain
{x+n-1}
}
{\right)}
}
}
@
wz≤r de Moivre'a
\EQEDhi
{\EQEDplain
{n}
}
{\EQEDbrackets
{\left[}
{\EQEDplain
{q}
\EQEDbrackets
{\left(}
{\cos \varphi \EQEDplain
{+}
\EQEDplain
{i}
\sin \varphi }
{\right)}
}
{\right]}
}
\EQEDplain
{=}
\EQEDhi
{\EQEDplain
{n}
}
{\varphi }
\EQEDbrackets
{\left(}
{\cos \EQEDplain
{n}
\varphi \EQEDplain
{+i}
\sin \EQEDplain
{n}
\varphi }
{\right)}
@
hiperboloida jednopow│okowa
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{x}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{a}
}
}
\EQEDplain
{+}
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{y}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{b}
}
}
\EQEDplain
{-}
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{z}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{c}
}
}
\EQEDplain
{=}
\EQEDplain
{1}
@
hiperboloida dwupow│okowa
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{x}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{a}
}
}
\EQEDplain
{+}
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{y}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{b}
}
}
\EQEDplain
{-}
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{z}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{c}
}
}
\EQEDplain
{=}
\EQEDplain
{-1}
@
sto┐ek
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{x}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{a}
}
}
\EQEDplain
{+}
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{y}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{b}
}
}
\EQEDplain
{-}
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{z}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{c}
}
}
\EQEDplain
{=}
\EQEDplain
{0}
@
paraboloida eliptyczna
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{x}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{a}
}
}
\EQEDplain
{+}
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{y}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{b}
}
}
\EQEDplain
{=}
\EQEDplain
{z}
@
paraboloida hiperboliczna
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{x}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{a}
}
}
\EQEDplain
{-}
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{y}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{b}
}
}
\EQEDplain
{=}
\EQEDplain
{z}
@
walec hiperboliczny
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{x}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{a}
}
}
\EQEDplain
{-}
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{y}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{b}
}
}
\EQEDplain
{=}
\EQEDplain
{1}
@
walec eliptyczny
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{x}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{a}
}
}
\EQEDplain
{+}
\frac
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{y}
}
}
{\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{b}
}
}
\EQEDplain
{=}
\EQEDplain
{1}
@
r≤wnanie og≤lne powierzchni stopnia II
\EQEDlo
{\EQEDplain
{11}
}
{\EQEDplain
{a}
}
\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{x}
}
\EQEDplain
{+}
\EQEDlo
{\EQEDplain
{22}
}
{\EQEDplain
{a}
}
\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{y}
}
\EQEDplain
{+}
\EQEDlo
{\EQEDplain
{33}
}
{\EQEDplain
{a}
}
\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{z}
}
\EQEDplain
{+2}
\EQEDlo
{\EQEDplain
{12}
}
{\EQEDplain
{a}
}
\EQEDplain
{xy}
\EQEDplain
{+}
\EQEDlo
{\EQEDplain
{23}
}
{\EQEDplain
{2a}
}
\EQEDplain
{yz}
\EQEDplain
{+2}
\EQEDlo
{\EQEDplain
{31}
}
{\EQEDplain
{a}
}
\EQEDplain
{zx}
\EQEDplain
{+}
\EQEDplain
{2}
\EQEDlo
{\EQEDplain
{14}
}
{\EQEDplain
{a}
}
\EQEDplain
{x+2}
\EQEDlo
{\EQEDplain
{24}
}
{\EQEDplain
{a}
}
\EQEDplain
{y+2}
\EQEDlo
{\EQEDplain
{34}
}
{\EQEDplain
{a}
}
\EQEDplain
{z+}
\EQEDlo
{\EQEDplain
{44}
}
{\EQEDplain
{a}
}
\EQEDplain
{=0}
@
r≤wnanie stycznej
\EQEDplain
{Y-y=}
\frac
{\EQEDplain
{dy}
}
{\EQEDplain
{dx}
}
\EQEDbrackets
{\left(}
{\EQEDplain
{X-x}
}
{\right)}
@
r≤wnanie normalnej
\EQEDplain
{Y-y=-}
\frac
{\EQEDplain
{1}
}
{\frac
{\EQEDplain
{y}
}
{\EQEDplain
{dx}
}
}
\EQEDbrackets
{\left(}
{\EQEDplain
{X-x}
}
{\right)}
@
pochodna
\EQEDhi
{\EQEDplain
{,}
}
{\EQEDplain
{f}
}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\EQEDplain
{=}
\EQEDlim
{\lim}
{\Delta \EQEDplain
{x}
\rightarrow \EQEDplain
{0}
}
{\frac
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDplain
{x+}
\Delta \EQEDplain
{x}
}
{\right)}
\EQEDplain
{-f}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
}
{\Delta \EQEDplain
{x}
}
}
@
pochodna cz╣stkowa
\frac
{\delta \EQEDplain
{u}
}
{\delta \EQEDplain
{x}
}
\EQEDplain
{=}
\EQEDlim
{\lim}
{\Delta \EQEDplain
{x}
\rightarrow \EQEDplain
{0}
}
{\frac
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDplain
{x+}
\Delta \EQEDplain
{x}
\EQEDplain
{,y,z,...,t}
}
{\right)}
\EQEDplain
{-f}
\EQEDbrackets
{\left(}
{\EQEDplain
{x,y,z,...,t}
}
{\right)}
}
{\Delta \EQEDplain
{x}
}
}
@
r≤┐niczka zupe│na
\EQEDplain
{du=}
\frac
{\delta \EQEDplain
{u}
}
{\delta \EQEDplain
{x}
}
\EQEDplain
{dx+}
\frac
{\delta \EQEDplain
{u}
}
{\delta \EQEDplain
{y}
}
\EQEDplain
{dy+...+}
\frac
{\delta \EQEDplain
{u}
}
{\delta \EQEDplain
{t}
}
\EQEDplain
{dt}
@
wz≤r Leibnitza
\EQEDhi
{\EQEDplain
{n}
}
{\EQEDplain
{D}
}
\EQEDbrackets
{\left(}
{\EQEDplain
{uv}
}
{\right)}
\EQEDplain
{=u}
\EQEDhi
{\EQEDplain
{n}
}
{\EQEDplain
{D}
}
\EQEDplain
{v+}
\EQEDchoose
{\EQEDplain
{n}
}
{\EQEDplain
{1}
}
\EQEDplain
{Du}
\EQEDhi
{\EQEDplain
{n-1}
}
{\EQEDplain
{D}
}
\EQEDplain
{v+}
\EQEDchoose
{\EQEDplain
{n}
}
{\EQEDplain
{2}
}
\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{D}
}
\EQEDplain
{u}
\EQEDhi
{\EQEDplain
{n-2}
}
{\EQEDplain
{D}
}
\EQEDplain
{v+...+}
\EQEDchoose
{\EQEDplain
{n}
}
{\EQEDplain
{k}
}
\EQEDhi
{\EQEDplain
{k}
}
{\EQEDplain
{D}
}
\EQEDplain
{u}
\EQEDhi
{\EQEDplain
{n-k}
}
{\EQEDplain
{D}
}
\EQEDplain
{v+...+}
\EQEDhi
{\EQEDplain
{n}
}
{\EQEDplain
{D}
}
\EQEDplain
{uv}
@
wz≤r prostok╣t≤w
\EQEDint
{\EQEDplain
{a}
}
{\EQEDplain
{b}
}
{\EQEDplain
{ydx}
\approx \EQEDplain
{h}
\EQEDbrackets
{\left(}
{\EQEDlo
{\EQEDplain
{0}
}
{\EQEDplain
{y}
}
\EQEDplain
{+}
\EQEDlo
{\EQEDplain
{1}
}
{\EQEDplain
{y}
}
\EQEDplain
{+...+}
\EQEDlo
{\EQEDplain
{n-1}
}
{\EQEDplain
{y}
}
}
{\right)}
}
@
wz≤r trapez≤w
\EQEDint
{\EQEDplain
{a}
}
{\EQEDplain
{b}
}
{\EQEDplain
{ydx}
\approx \frac
{\EQEDplain
{1}
}
{\EQEDplain
{2}
}
\EQEDplain
{h}
\EQEDbrackets
{\left(}
{\EQEDlo
{\EQEDplain
{0}
}
{\EQEDplain
{y}
}
\EQEDplain
{+2}
\EQEDlo
{\EQEDplain
{1}
}
{\EQEDplain
{y}
}
\EQEDplain
{+2}
\EQEDlo
{\EQEDplain
{2}
}
{\EQEDplain
{y}
}
\EQEDplain
{+...+2}
\EQEDlo
{\EQEDplain
{n-1}
}
{\EQEDplain
{y}
}
\EQEDplain
{+}
\EQEDlo
{\EQEDplain
{n}
}
{\EQEDplain
{y}
}
}
{\right)}
}
@
wz≤r parabol (Simpsona)
\EQEDint
{\EQEDplain
{a}
}
{\EQEDplain
{b}
}
{\EQEDplain
{ydx}
\approx \frac
{\EQEDplain
{1}
}
{\EQEDplain
{3}
}
\EQEDplain
{h}
\EQEDbrackets
{\left(}
{\EQEDlo
{\EQEDplain
{0}
}
{\EQEDplain
{y}
}
\EQEDplain
{+4}
\EQEDlo
{\EQEDplain
{1}
}
{\EQEDplain
{y}
}
\EQEDplain
{+2}
\EQEDlo
{\EQEDplain
{2}
}
{\EQEDplain
{y}
}
\EQEDplain
{+4}
\EQEDlo
{\EQEDplain
{3}
}
{\EQEDplain
{y}
}
\EQEDplain
{+...+2}
\EQEDlo
{\EQEDplain
{n-2}
}
{\EQEDplain
{y}
}
\EQEDplain
{+4}
\EQEDlo
{\EQEDplain
{n-1}
}
{\EQEDplain
{y}
}
\EQEDplain
{+}
\EQEDlo
{\EQEDplain
{n}
}
{\EQEDplain
{y}
}
}
{\right)}
}
@
ca│ka niew│a£ciwa
\EQEDint
{\EQEDplain
{a}
}
{\infty }
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\EQEDplain
{dx=}
\EQEDlim
{\lim}
{\EQEDplain
{B}
\rightarrow \infty }
{\EQEDint
{\EQEDplain
{a}
}
{\EQEDplain
{B}
}
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\EQEDplain
{dx}
}
}
}
@
ca│ka krzywoliniowa
\EQEDint
{\EQEDplain
{K}
}
{\EQEDplain
{}
}
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDplain
{x,y}
}
{\right)}
\EQEDplain
{ds=}
\EQEDlim
{\lim}
{\Delta \EQEDlo
{\EQEDplain
{i}
}
{\EQEDplain
{s}
}
\rightarrow \EQEDplain
{0,n}
\rightarrow \infty }
{\EQEDsum
{\EQEDplain
{i=1}
}
{\EQEDplain
{n}
}
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDlo
{\EQEDplain
{i}
}
{\xi }
\EQEDplain
{,}
\EQEDlo
{\EQEDplain
{i}
}
{\eta }
}
{\right)}
\Delta \EQEDlo
{\EQEDplain
{i-1}
}
{\EQEDplain
{s}
}
}
}
}
@
ca│ka podw≤jna
\EQEDint
{\EQEDplain
{S}
}
{\EQEDplain
{}
}
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDplain
{x,y}
}
{\right)}
\EQEDplain
{dS=}
\EQEDlim
{\lim}
{\Delta \EQEDlo
{\EQEDplain
{i}
}
{\EQEDplain
{S}
}
\rightarrow \EQEDplain
{0,n}
\rightarrow \infty }
{\EQEDsum
{\EQEDplain
{i=1}
}
{\EQEDplain
{n}
}
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDlo
{\EQEDplain
{i}
}
{\EQEDplain
{x}
}
\EQEDplain
{,}
\EQEDlo
{\EQEDplain
{i}
}
{\EQEDplain
{y}
}
}
{\right)}
\EQEDplain
{d}
\EQEDlo
{\EQEDplain
{i}
}
{\EQEDplain
{S}
}
}
}
}
@
ca│ka potr≤jna
\EQEDint
{\EQEDplain
{V}
}
{\EQEDplain
{}
}
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDplain
{x,y,z}
}
{\right)}
\EQEDplain
{dV=}
\EQEDlim
{\lim}
{\Delta \EQEDlo
{\EQEDplain
{i}
}
{\EQEDplain
{V}
}
\rightarrow \EQEDplain
{0,n}
\rightarrow \infty }
{\EQEDsum
{\EQEDplain
{i=1}
}
{\EQEDplain
{n}
}
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\EQEDlo
{\EQEDplain
{i}
}
{\EQEDplain
{x}
}
\EQEDplain
{,}
\EQEDlo
{\EQEDplain
{i}
}
{\EQEDplain
{y}
}
\EQEDplain
{,}
\EQEDlo
{\EQEDplain
{i}
}
{\EQEDplain
{z}
}
}
{\right)}
\EQEDplain
{d}
\EQEDlo
{\EQEDplain
{i}
}
{\EQEDplain
{V}
}
}
}
}
@
wz≤r Stokesa
\EQEDint
{\EQEDplain
{K}
}
{\EQEDplain
{}
}
{\EQEDplain
{Pdx+Qdy+Rdz=}
\EQEDint
{\EQEDplain
{S}
}
{\EQEDplain
{}
}
{\EQEDbrackets
{\left(}
{\frac
{\delta \EQEDplain
{Q}
}
{\delta \EQEDplain
{x}
}
\EQEDplain
{-}
\frac
{\delta \EQEDplain
{P}
}
{\delta \EQEDplain
{y}
}
}
{\right)}
\EQEDplain
{dxdy+}
\EQEDbrackets
{\left(}
{\frac
{\delta \EQEDplain
{R}
}
{\delta \EQEDplain
{y}
}
\EQEDplain
{}
\frac
{\delta \EQEDplain
{Q}
}
{\delta \EQEDplain
{z}
}
}
{\right)}
\EQEDplain
{dydz+}
\EQEDbrackets
{\left(}
{\frac
{\delta \EQEDplain
{P}
}
{\delta \EQEDplain
{z}
}
\EQEDplain
{}
\frac
{\delta \EQEDplain
{R}
}
{\delta \EQEDplain
{x}
}
}
{\right)}
\EQEDplain
{dzdx}
}
}
@
wz≤r Ostrogradzkiego-Gaussa
\EQEDint
{\EQEDplain
{}
}
{\EQEDplain
{}
}
{\EQEDint
{\EQEDplain
{V}
}
{\EQEDplain
{}
}
{\EQEDint
{\EQEDplain
{}
}
{\EQEDplain
{}
}
{\EQEDbrackets
{\left(}
{\frac
{\delta \EQEDplain
{P}
}
{\delta \EQEDplain
{x}
}
\EQEDplain
{+=}
\frac
{\delta \EQEDplain
{Q}
}
{\delta \EQEDplain
{y}
}
\EQEDplain
{+}
\frac
{\delta \EQEDplain
{R}
}
{\delta \EQEDplain
{z}
}
}
{\right)}
\EQEDplain
{dV=}
\EQEDint
{\EQEDplain
{}
}
{\EQEDplain
{}
}
{\EQEDint
{\EQEDplain
{S}
}
{\EQEDplain
{}
}
{\EQEDplain
{Pdydz+Qdzdx+Rdxdy}
}
}
}
}
}
@
r≤wnanie Bernoulliego
\EQEDhi
{\EQEDplain
{,}
}
{\EQEDplain
{y}
}
\EQEDplain
{+P}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\EQEDplain
{y=Q}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\EQEDhi
{\EQEDplain
{n}
}
{\EQEDplain
{y}
}
@
r≤wnanie Riccatiego
\EQEDhi
{\EQEDplain
{,}
}
{\EQEDplain
{y}
}
\EQEDplain
{=P}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\EQEDhi
{\EQEDplain
{2}
}
{\EQEDplain
{y}
}
\EQEDplain
{+Q}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
\EQEDplain
{y+R}
\EQEDbrackets
{\left(}
{\EQEDplain
{x}
}
{\right)}
@
r≤wnanie Lagrange'a
\EQEDplain
{a}
\EQEDbrackets
{\left(}
{\EQEDhi
{\EQEDplain
{,}
}
{\EQEDplain
{y}
}
}
{\right)}
\EQEDplain
{x+b}
\EQEDbrackets
{\left(}
{\EQEDhi
{\EQEDplain
{,}
}
{\EQEDplain
{y}
}
}
{\right)}
\EQEDplain
{y+c}
\EQEDbrackets
{\left(}
{\EQEDhi
{\EQEDplain
{,}
}
{\EQEDplain
{y}
}
}
{\right)}
\EQEDplain
{=0}
@
wz≤r Eulera 1
\sin \EQEDplain
{z=}
\frac
{\EQEDhi
{\EQEDplain
{iz}
}
{\EQEDplain
{e}
}
\EQEDplain
{-}
\EQEDhi
{\EQEDplain
{-iz}
}
{\EQEDplain
{e}
}
}
{\EQEDplain
{2i}
}
@
wz≤r Eulera 2
\cos \EQEDplain
{z=}
\frac
{\EQEDhi
{\EQEDplain
{iz}
}
{\EQEDplain
{e}
}
\EQEDplain
{+}
\EQEDhi
{\EQEDplain
{-iz}
}
{\EQEDplain
{e}
}
}
{\EQEDplain
{2}
}
@
wz≤r ca│kowy Cauchy'ego
\EQEDhi
{\EQEDbrackets
{\left(}
{\EQEDplain
{n}
}
{\right)}
}
{\EQEDplain
{f}
}
\EQEDbrackets
{\left(}
{\EQEDplain
{z}
}
{\right)}
\EQEDplain
{=}
\frac
{\EQEDplain
{n!}
}
{\EQEDplain
{2}
\Pi \EQEDplain
{i}
}
\EQEDint
{\EQEDplain
{C}
}
{\EQEDplain
{}
}
{\frac
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\zeta }
{\right)}
}
{\EQEDhi
{\EQEDplain
{n+1}
}
{\EQEDbrackets
{\left(}
{\zeta \EQEDplain
{-z}
}
{\right)}
}
}
\EQEDplain
{d}
\zeta }
@
residua
\EQEDplain
{resf}
\EQEDlo
{\EQEDplain
{z=a}
}
{\EQEDbrackets
{\left(}
{\EQEDplain
{z}
}
{\right)}
}
\EQEDplain
{=}
\frac
{\EQEDplain
{1}
}
{\EQEDplain
{2}
\Pi \EQEDplain
{i}
}
\EQEDint
{\EQEDplain
{C}
}
{\EQEDplain
{}
}
{\EQEDplain
{f}
\EQEDbrackets
{\left(}
{\zeta }
{\right)}
\EQEDplain
{d}
\zeta }
@
wz≤r Bayesa
\EQEDplain
{P}
\EQEDbrackets
{\left(}
{\EQEDlo
{\EQEDplain
{i}
}
{\EQEDplain
{A}
}
\EQEDbrackets
{\left\|}
{\EQEDplain
{B}
}
{\right.}
}
{\right)}
\EQEDplain
{=}
\frac
{\EQEDplain
{P}
\EQEDbrackets
{\left(}
{\EQEDplain
{B}
\EQEDbrackets
{\left\|}
{\EQEDlo
{\EQEDplain
{i}
}
{\EQEDplain
{A}
}
}
{\right.}
}
{\right)}
\EQEDplain
{P}
\EQEDbrackets
{\left(}
{\EQEDlo
{\EQEDplain
{i}
}
{\EQEDplain
{A}
}
}
{\right)}
}
{\EQEDsum
{\EQEDplain
{j=1}
}
{\EQEDplain
{N}
}
{\EQEDplain
{P}
\EQEDbrackets
{\left(}
{\EQEDplain
{B}
\EQEDbrackets
{\left\|}
{\EQEDlo
{\EQEDplain
{j}
}
{\EQEDplain
{A}
}
}
{\right.}
}
{\right)}
\EQEDplain
{P}
\EQEDbrackets
{\left(}
{\EQEDlo
{\EQEDplain
{j}
}
{\EQEDplain
{A}
}
}
{\right)}
}
}
@
rozk│ad Bernoulliego
\EQEDplain
{P}
\EQEDbrackets
{\left(}
{\EQEDplain
{A}
}
{\right)}
\EQEDplain
{=}
\EQEDchoose
{\EQEDplain
{n}
}
{\EQEDplain
{k}
}
\EQEDhi
{\EQEDplain
{k}
}
{\EQEDplain
{p}
}
\EQEDhi
{\EQEDplain
{n-k}
}
{\EQEDbrackets
{\left(}
{\EQEDplain
{1-p}
}
{\right)}
}
@