উইকিসংকলন

সাহায্য:Fractions and functions

← ← Help:Contents ভগ্নাংশ এবং ফাংশন
গাণিতিক ভগ্নাংশ এবং ফাংশন প্রদর্শনের নির্দেশিকা

ভগ্নাংশ

সম্পাদনা

ভগ্নাংশ ক্ষেত্রবিশেষে সাধারণ পাঠ্যতে আসতে পারে, আর তখন সঠিকভাবে প্রদর্শন প্রয়োজন হয়। UTF-8 এর সংগ্রহে খুবই সীমিত কয়েকটি দেয়া আছে: এগুলো হলো ¼ ½ ¾ ⅓ ⅔ ⅕ ⅖ ⅗ ⅘ ⅙ ⅚ ⅛ ⅜ ⅝ ⅞

সম্পাদনা পাতার নিচের দিকে এগুলোর বেশ কিছু ভগ্নাংশ যোগ করা আছে। বাকিগুলো দেয়া হয়নি কেননা সেগুলো উইকিসংকলনে ব্যবহৃত ফন্টে ফুটে ওঠে না। এগুলো এখানে দেখাবে যদি আপনার কম্পিউটারে Lucida Sans Unicode থেকে থাকে। এরা সাধারণ ফন্ট সাইজেে কিছুটা বড় দেখতে।

অন্যান্য ভগ্নাংশগুলো কিবোর্ডের অক্ষরগুলোর দ্বারা প্রকাশ করা যায় (যেমন ১১/১৬, ২৭/৩২, প্রভৃতি), তবে সেগুলো দেখতে খটোমটো বা বড় দেখায়। HTML কোডিং ব্যবহার করে বেশ সুন্দরভাবে কিছু ভগ্নাংশ দেখানো যায় যেমন superscript/subscript মার্কআপ দিয়ে।

তাহলে, ১১/১৬ দেখাতে হবে <span style="vertical-align:super;font-size:x-small">১১</span>/<span style="vertical-align:sub;font-size:x-small">১৬</span>

Template:sfrac ব্যবহার করে উল্লম্বভাবেই ভগ্নাংশ প্রকাশ করা যায় মিডিয়াউইকি টাইপফেসে। উদা। ৪ ৩/৪, ৬৭/৮৩৭২৬।

ভগ্নাংশ যদিও TeX দিয়ে বানানো যায়, সেগুলো মিডিয়াউইকি টাইপফেসের সাথে মেলে না।

ফাংশন

সম্পাদনা

মিডিয়াউইকিতে ফাংশন দেখানো হয় TeX এর একটি সংস্করণের মাধ্যমে। পরের পাঠ্যগুলো Meta-Wiki থেকে আনা।

TeX

সম্পাদনা

w:MediaWiki তে TeX মার্কআপ এর একটি সাবসেট ব্যবহার করা হয় গাণিতিক সংকেত দেখাতে। meta:MediaWiki math markup দেখুন।

এটি হয় PNG ছবি অথবা স্রেফ HTML মার্কআপ তৈরি করে, যা ব্যবহারকারীর পছন্দ এবং গাণিতিক প্রকাশের জটিলতার উপর নির্ভর করে। ভবিষ্যতে, যখন আরও সকল ব্রাউজার সক্ষম হবে, তখন বিশেষ ধরনের HTML বা এমনকি MathML দিয়ে প্রকাশিত হবে অনেকক্ষেত্রেই।

(More precisely, w:MediaWiki filters the markup through Texvc, which in turn passes the commands to TeX for the actual rendering. Thus, only a limited part of the full w:TeX language is supported; see below for details.)

Math markup goes inside <math> ... </math>. The edit toolbar has a button for this.

MediaWiki templates, variables and parameters cannot be used within math tags, see Demo of attempt to use parameters within TeX.

The PNG images are black on white (not transparent). These colors, as well as font sizes and types, are independent of browser settings or css. Font sizes and types will often deviate from what HTML renders. The css selector of the images is img.tex.

In the case of a non-white page background, the white background of the formula effectively highlights it, which can be an advantage or a disadvantage.

One may want to avoid using TeX markup as part of a line of regular text, as the formulae don't align properly and the font size, as said, usually does not match.

The alt attribute of the TeX images (the text that is displayed if your browser can't display images; Internet Explorer even shows it up in the hover box) is the wikitext that produced them, excluding the <math> and </math>.

Discussion, bug reports and feature requests should go to the Wikitech-l mailing list. These can also be filed on Mediazilla under MediaWiki extensions.

General

সম্পাদনা

Spaces and newlines are mostly ignored. Apart from function and operator names, as is customary in mathematics for variables, letters are in italics; digits are not. For other text, (like variable labels) to avoid being rendered in italics like variables, use \mbox or \mathrm: <math>\mbox{abc}</math> gives  

Line breaks help keep the wikitext clear, for instance, a line break after each term or matrix row.

Size

সম্পাদনা

There are some possibilities, to change the size of the formulas. For example, fractions can be made smaller using \tfrac instead of \frac.

  becomes  .

In general, formulas can be made even smaller if \scriptstyle is employed:

  becomes  .

Functions, symbols, special characters

সম্পাদনা

For producing special characters without math tags, see m:Help:Special characters.

Comparison:

  • &alpha; gives "α"
    • <math>\alpha</math> gives  ,
    • ("&" and ";" vs. "\", in this case the same code word "alpha");
  • &radic;2 gives "√2"
    • <math>\sqrt{2}</math> gives  
    • (the same difference as above, but also another code word, "radic" vs. "sqrt"; in TeX braces);
  • &radic;(1-''e''&sup2;) gives √(1-e²),
    • <math>\sqrt{1-e^2}</math> gives  ,
    • (parentheses vs. braces, "''e''" vs. "e", "&sup2;" vs. "^2").
Feature Syntax How it looks rendered
Accents/Diacritics \acute{a} \quad \grave{a} \quad \breve{a} \quad \check{a} \quad \tilde{a}  
Std. functions (good) \sin x + \ln y +\operatorname{sgn} z

\sin a \ \cos b \ \tan c \ \cot d \ \sec e \ \csc f
\sinh g \ \cosh h \ \tanh i \ \coth j
\arcsin k \ \arccos l \ \arctan m
\lim n \ \limsup o \ \liminf p
\min q \ \max r \ \inf s \ \sup t
\exp u \ \lg v \ \log w
\ker x \ \deg x \gcd x \Pr x \ \det x \hom x \ \arg x \dim x

 

 
 
 
 
 
 
 

Std. functions (wrong) sin x + ln y + sgn z  
Modular arithmetic s_k \equiv 0 \pmod{m}

a \bmod b

 

 

Derivatives \nabla \; \partial x \; dx \; \dot x \; \ddot y  
Sets

(Square symbols may not work for some wikis)

\forall \; \exists \; \empty \; \emptyset \; \varnothing \in \ni \not\in \notin

\subset \subseteq \supset \supseteq \cap \bigcap \cup \bigcup \biguplus

 

 

\sqsubset \sqsubseteq \sqsupset \sqsupseteq \sqcap \sqcup \bigsqcup  
Logic p \land \wedge \; \bigwedge \; \bar{q} \to p \; \lor \vee \; \bigvee \; \lnot \; \neg q \; \setminus \; \smallsetminus  
Root \sqrt{2}\approx 1.4  
\sqrt[n]{x}  
Relations \sim \; \approx \; \simeq \; \cong \; \le \; < \; \ll \; \gg \; \ge \; > \; \equiv \; \not\equiv \; \ne \; \propto \; \pm \; \mp  
Geometric \Diamond \; \Box \; \triangle \; \angle \; \perp \; \mid \; \nmid \; \| \; 45^\circ  
Arrows

(Harpoons may not work for some wikis)

\leftarrow \; \gets \; \rightarrow \; \to \; \leftrightarrow

\longleftarrow \; \longrightarrow
\mapsto \; \longmapsto \; \hookrightarrow \; \hookleftarrow
\nearrow \; \searrow \; \swarrow \; \nwarrow
\uparrow \; \downarrow \; \updownarrow

 

 
 
 
 

\rightharpoonup \; \rightharpoondown \; \leftharpoonup \; \leftharpoondown \; \upharpoonleft \; \upharpoonright \; \downharpoonleft \; \downharpoonright  
\Leftarrow \; \Rightarrow \; \Leftrightarrow

\Longleftarrow \; \Longrightarrow \; \Longleftrightarrow (or \iff)
\Uparrow \; \Downarrow \; \Updownarrow

 

 
 

Special \And; \eth \; \S \; \P \; \% \; \dagger \; \ddagger \; \star \; * \; \ldots

\smile \frown \wr \oplus \bigoplus \otimes \bigotimes
\times \cdot \circ \bullet \bigodot \triangleleft \triangleright \infty \bot \top \vdash \vDash \Vdash \models \lVert \rVert
\imath \; \hbar \; \ell \; \mho \; \Finv \; \Re \; \Im \; \wp \; \complement \quad \diamondsuit \; \heartsuit \; \clubsuit \; \spadesuit \; \Game \quad \flat \; \natural \; \sharp

 

 
 
 

Lowercase \mathcal has some extras \mathcal {45abcdenpqstuvwx}  

Subscripts, superscripts, integrals

সম্পাদনা
Feature Syntax How it looks rendered
HTML PNG
Superscript a^2    
Subscript a_2    
Grouping a^{2+2}    
a_{i,j}    
Combining sub & super x_2^3  
Preceding sub & super {}_1^2\!X_3^4  
Derivative (good) x', y''    
Derivative (wrong in HTML) x^\prime, y^{\prime\prime}    
Derivative (wrong in PNG) x\prime, y\prime\prime    
Derivative dots \dot{x}, \ddot{x}  
Underlines, overlines, vectors \hat a \ \bar b \ \vec c \ \overrightarrow{a b} \ \overleftarrow{c d} \ \widehat{d e f} \ \overline{g h i} \ \underline{j k l}  
Overbraces

\begin{matrix} 5050 \\ \overbrace{ 1+2+\cdots+100 } \end{matrix}

 

Underbraces

\begin{matrix} \underbrace{ a+b+\cdots+z } \\ 26 \end{matrix}

 

Sum \sum_{k=1}^N k^2  
Product \prod_{i=1}^N x_i  
Coproduct \coprod_{i=1}^N x_i  
Limit \lim_{n \to \infty}x_n  
Integral \int_{-N}^{N} e^x\, dx  
Double integral \iint_{D}^{W} \, dx\,dy  
Triple integral \iiint_{E}^{V} \, dx\,dy\,dz  
Quadruple integral \iiiint_{F}^{U} \, dx\,dy\,dz\,dt  
Path integral \oint_{C} x^3\, dx + 4y^2\, dy  
Intersections \bigcap_1^{n} p  
Unions \bigcup_1^{k} p  

Fractions, matrices, multilines

সম্পাদনা
Feature Syntax How it looks rendered
Fractions \frac{2}{4} or {2 \over 4}  
Binomial coefficients {n \choose k}  
Small Fractions \begin{matrix} \frac{2}{4} \end{matrix}  
Matrices \begin{matrix} x & y \\ z & v \end{matrix}  
\begin{vmatrix} x & y \\ z & v \end{vmatrix}  
\begin{Vmatrix} x & y \\ z & v \end{Vmatrix}  
\begin{bmatrix} 0 & \cdots & 0 \\ \vdots &

\ddots & \vdots \\ 0 & \cdots &

0\end{bmatrix}		  
\begin{Bmatrix} x & y \\ z & v \end{Bmatrix}  
\begin{pmatrix} x & y \\ z & v \end{pmatrix}  
Case distinctions f(n) = \begin{cases} n/2, & \mbox{if }n\mbox{ is even} \\ 3n+1, & \mbox{if }n\mbox{ is odd} \end{cases}  
Multiline equations \begin{matrix}f(n+1) & = & (n+1)^2 \\ \ & = & n^2 + 2n + 1 \end{matrix}  
Alternative multiline equations (using tables) 

{|
|-
|<math>f(n+1)</math>
|<math>=(n+1)^2</math>
|-
|
|<math>=n^2 + 2n + 1</math>
|}
   
 

Fonts

সম্পাদনা
Feature Syntax How it looks rendered

(Note the lack of omicron; note also that several upper case Greek letters are rendered identically to the corresponding Roman ones)

\Alpha\ \Beta\ \Gamma\ \Delta\ \Epsilon\ \Zeta\ \Eta\ \Theta\ \Iota\ \Kappa\ \Lambda\ \Mu\ \Nu\ \Xi\ \Pi\ \Rho\ \Sigma\ \Tau\ \Upsilon\ \Phi\ \Chi\ \Psi\ \Omega

\alpha\ \beta\ \gamma\ \delta\ \epsilon\ \zeta\ \eta\ \theta\ \iota\ \kappa\ \lambda\ \mu\ \nu\ \xi\ \pi\ \rho\ \sigma\ \tau\ \upsilon\ \phi\ \chi\ \psi\ \omega

\varepsilon\ \digamma\ \vartheta\ \varkappa\ \varpi\ \varrho\ \varsigma\ \varphi

 

 

 

\mathbb{N}\ \mathbb{Z}\ \mathbb{Q}\ \mathbb{R}\ \mathbb{C}  
(vectors) \mathbf{x}\cdot\mathbf{y} = 0  
boldface (greek) \boldsymbol{\alpha} + \boldsymbol{\beta} + \boldsymbol{\gamma}  
italics \mathit{ABCDE abcde 1234}  
\mathrm{ABCDE abcde 1234}  
\mathfrak{ABCDE abcde 1234}  
Calligraphy/Script \mathcal{ABCDE abcde 1234}  
\aleph \beth \gimel \daleth  
non-italicised characters \mbox{abc}    
mixed italics (bad) \mbox{if} n \mbox{is even}    
mixed italics (good) \mbox{if }n\mbox{ is even}    

Parenthesizing big expressions, brackets, bars

সম্পাদনা
Feature Syntax How it looks rendered
Not good ( \frac{1}{2} )  
Better \left ( \frac{1}{2} \right )  

You can use various delimiters with \left and \right:

Feature Syntax How it looks rendered
Parentheses \left ( \frac{a}{b} \right )  
Brackets \left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack  
Braces \left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace  
Angle brackets \left \langle \frac{a}{b} \right \rangle  
Bars and double bars \left | \frac{a}{b} \right \vert \left \Vert \frac{c}{d} \right \|  
Floor and ceiling functions: \left \lfloor \frac{a}{b} \right \rfloor \left \lceil \frac{c}{d} \right \rceil  
Slashes and backslashes \left / \frac{a}{b} \right \backslash  
Up, down and up-down arrows \left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow  

Delimiters can be mixed,
as long as \left and \right match

\left [ 0,1 \right )
\left \langle \psi \right |

 
 

Use \left. and \right. if you don't
want a delimiter to appear:		 \left . \frac{A}{B} \right \} \to X  
Size of the delimiters \big( \Big( \bigg( \Bigg( ... \Bigg] \bigg] \Big] \big]

 

\big\{ \Big\{ \bigg\{ \Bigg\{ ... \Bigg\rangle \bigg\rangle \Big\rangle \big\rangle

 

\big\| \Big\| \bigg\| \Bigg\| ... \Bigg| \bigg| \Big| \big|  
\big\lfloor \Big\lfloor \bigg\lfloor \Bigg\lfloor ... \Bigg\rceil \bigg\rceil \Big\rceil \big\rceil

 

\big\uparrow \Big\uparrow \bigg\uparrow \Bigg\uparrow ... \Bigg\Downarrow \bigg\Downarrow \Big\Downarrow \big\Downarrow

 

Spacing

সম্পাদনা

Note that TeX automatically handles most spacing, but you may sometimes want manual control.

Feature Syntax How it looks rendered
double quad space a \qquad b  
quad space a \quad b  
text space a\ b  
text space without PNG conversion a \mbox{ } b  
large space a\;b  
medium space a\>b [not supported]
small space a\,b  
no space ab  
negative space a\!b  

Align with normal text flow

সম্পাদনা

Due to the default css 

img.tex { vertical-align: middle; }

an inline expression like   should look good.

If you need to align it otherwise, use <font style="vertical-align:-100%;"><math>...</math></font> and play with the vertical-align argument until you get it right; however, how it looks may depend on the browser and the browser settings.

Forced PNG rendering

সম্পাদনা

To force the formula to render as PNG, add \, (small space) at the end of the formula (where it is not rendered). This will force PNG if the user is in "HTML if simple" mode, but not for "HTML if possible" mode (math rendering settings in preferences).

You can also use \,\! (small space and negative space, which cancel out) anywhere inside the math tags. This does force PNG even in "HTML if possible" mode, unlike \,.

This could be useful to keep the rendering of formulae in a proof consistent, for example, or to fix formulae that render incorrectly in HTML (at one time, a^{2+2} rendered with an extra underscore), or to demonstrate how something is rendered when it would normally show up as HTML (as in the examples above).

For instance:

Syntax How it looks rendered
a^{c+2}  
a^{c+2} \,  
a^{\,\!c+2}  
a^{b^{c+2}}   (WRONG with option "HTML if possible or else PNG"!)
a^{b^{c+2}} \,   (WRONG with option "HTML if possible or else PNG"!)
a^{b^{c+2}}\approx 5   (due to " " correctly displayed, no code "\,\!" needed)
a^{b^{\,\!c+2}}  
\int_{-N}^{N} e^x\, dx  
\int_{-N}^{N} e^x\, dx \,  
\int_{-N}^{N} e^x\, dx \,\!  

This has been tested with most of the formulae on this page, and seems to work perfectly.

You might want to include a comment in the HTML so people don't "correct" the formula by removing it:

<!-- The \,\! is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->

Additional help sources

সম্পাদনা

Examples

সম্পাদনা

Quadratic Polynomial

সম্পাদনা 
 

<math>x_1 = a^2 + b^2 + c^2 </math>

Quadratic Formula

সম্পাদনা 
 

<math>x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}</math>

Parentheses and fractions

সম্পাদনা 
 

<math>\left(3-x\right) \times \left( \frac{2}{3-x} \right) =
\left(3-x\right) \times \left( \frac{3}{2-x} \right)</math>

Tall Parentheses and fractions

সম্পাদনা 
 

<math>2 = \left( \frac{\left(3-x\right) \times 3}{2-x} \right)</math>

Force Rendering

সম্পাদনা 
 

<math>4-2x = 9-3x \!</math>

Force Rendering

সম্পাদনা 
 

<math>-2x+3x = 9-4 \!</math>

Integrals

সম্পাদনা 
 

<math>\int_a^x \int_a^s f(y)\,dy\,ds = \int_a^x f(y)(x-y)\,dy\,</math>

Summation

সম্পাদনা 
 

<math>\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{m^2\,n}
{3^m\left(m\,3^n+n\,3^m\right)}</math>

Differential Equation

সম্পাদনা 
 

<math>u'' + p(x)u' + q(x)u=f(x),\,\,\,x>a</math>

Example

সম্পাদনা 
 

<math>|\bar{z}| = |z|, |(\bar{z})^n| = |z|^n, \arg(z^n) = n \arg(z)\,</math>

Limits

সম্পাদনা 
 

<math>\lim_{z\rightarrow z_0} f(z)=f(z_0)\,</math>

Integral Equation

সম্পাদনা 
 

<math>\phi_n(\kappa) = \frac{1}{4\pi^2\kappa^2} \int_0^\infty
\frac{\sin(\kappa R)}{\kappa R} \frac{\partial}{\partial R}\left[R^2\frac{\partial
D_n(R)}{\partial R}\right]\,dR\,</math>

Example

সম্পাদনা 
 
 
<math>\int_0^\infty x^\alpha \sin(x)\,dx = 2^\alpha \sqrt{\pi}\,
\frac{\Gamma(\frac{\alpha}{2}+1)}{\Gamma(\frac{1}{2}-\frac{\alpha}{2})}\,</math>

Example

সম্পাদনা 
 

<math>\phi_n(\kappa) = 
0.033C_n^2\kappa^{-11/3},\,\,\,\frac{1}{L_0}<\!\!<\kappa<\!\!<\frac{1}{l_0}\,</math>

Example

সম্পাদনা 
 

<math>f(x) = {a_0\over 2} + \sum_{n=1}^\infty a_n\cos\left({2n\pi x \over T}\right) +
b_n\sin\left({2n\pi x\over T}\right)\,</math>

Continuation and cases

সম্পাদনা 
 

f(x) = \begin{cases}1 & -1 \le x < 0\\
\frac{1}{2} & x = 0\\x&0<x\le 1\end{cases}

Example

সম্পাদনা 
 

<math>\Gamma(z) = \int_0^\infty e^{-t} t^{z-1} \,dt\,</math>

Example

সম্পাদনা 
 

<math>J_p(z) = \sum_{k=0}^\infty
\frac{(-1)^k\left(\frac{z}{2}\right)^{2k+p}}{k!\Gamma(k+p+1)}\,</math>

Example

সম্পাদনা 
 

 <math>{}_pF_q(a_1,...,a_p;c_1,...,c_q;z) = \sum_{n=0}^\infty
\frac{(a_1)_n\cdot\cdot\cdot(a_p)_n}{(c_1)_n\cdot\cdot\cdot(c_q)_n}\frac{z^n}{n!}\,</math>

Gamma Function

সম্পাদনা 
 

<math>\Gamma(n+1) = n \Gamma(n),  \; n>0</math>

Example

সম্পাদনা 
<math>\int_0^1 \frac{1}{\sqrt{-\ln x}} dx\,</math>

 

Example

সম্পাদনা 
 

<math>\int_0^\infty e^{-st}t^{x-1}\,dt,\ s>0\,</math>

Example

সম্পাদনা 
 

<math>B(u) = \sum_{k=0}^N {P_k}{N! \over k!(N - k)!}{u^k}(1 - 
u)^{N-k}\,</math>

Example

সম্পাদনা 
 

<math>u(x,y) = \frac{1}{\sqrt{2\pi}}\int_0^\infty 
f(\xi)\left[g(|x+\xi|,y)+g(|x-\xi|,y)\right]\,d\xi\,</math>

See also

সম্পাদনা
সম্পাদনা
মুদ্রণ সংশোধন করা
সারসংক্ষেপ
প্রাথমিক সাহায্য
বিশেষ সাহায্য
সূত্র
সহযোগিতা
'https://bn.wikisource.org/w/index.php?title=সাহায্য:Fractions_and_functions&oldid=1752128' থেকে আনীত