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1.1     ! root        1: .\"    @(#)g1  6.1 (Berkeley) 5/22/86
        !             2: .\"
        !             3: .if t .2C
        !             4: .SC Introduction
        !             5: .PP
        !             6: .UC EQN
        !             7: is a
        !             8: program for typesetting mathematics
        !             9: on the Graphics Systems phototypesetters on the
        !            10: .UX
        !            11: operating system.
        !            12: The 
        !            13: .UC EQN
        !            14: language was designed to be easy to use
        !            15: by people who know neither mathematics
        !            16: nor typesetting.
        !            17: Thus
        !            18: .UC EQN
        !            19: knows relatively little about mathematics.
        !            20: In particular, mathematical symbols like
        !            21: +, \(mi, \(mu, parentheses, and so on have no special meanings.
        !            22: .UC EQN
        !            23: is quite happy to set garbage (but it will look good).
        !            24: .PP
        !            25: .UC EQN
        !            26: works as a preprocessor for the typesetter formatter,
        !            27: .UC TROFF [1],
        !            28: so the normal mode of operation is to prepare
        !            29: a document with both mathematics and ordinary text
        !            30: interspersed,
        !            31: and let
        !            32: .UC EQN
        !            33: set the mathematics while
        !            34: .UC TROFF
        !            35: does the body of the text.
        !            36: .PP
        !            37: On
        !            38: .UC UNIX ,
        !            39: .UC EQN
        !            40: will also produce mathematics on
        !            41: .UC DASI 
        !            42: and
        !            43: .UC GSI
        !            44: terminals and on
        !            45: Model 37 teletypes.
        !            46: The input is identical, but you have to use the programs
        !            47: .UC NEQN 
        !            48: and
        !            49: .UC NROFF
        !            50: instead of
        !            51: .UC EQN 
        !            52: and
        !            53: .UC TROFF .
        !            54: Of course, some things won't look as good
        !            55: because terminals 
        !            56: don't provide the variety of characters, sizes and fonts
        !            57: that a typesetter does,
        !            58: but the output is usually adequate for proofreading.
        !            59: .PP
        !            60: To use 
        !            61: .UC EQN
        !            62: on
        !            63: .UC UNIX ,
        !            64: .P1
        !            65: eqn files | troff
        !            66: .P2
        !            67: .SC Displayed Equations
        !            68: .PP
        !            69: To tell
        !            70: .UC EQN
        !            71: where a mathematical expression begins and ends,
        !            72: we mark it with lines beginning
        !            73: .UC .EQ
        !            74: and
        !            75: .UC .EN .
        !            76: Thus
        !            77: if you type the lines
        !            78: .P1
        !            79: ^EQ
        !            80: x=y+z
        !            81: ^EN
        !            82: .P2
        !            83: your output will look like
        !            84: .EQ
        !            85: x=y+z
        !            86: .EN
        !            87: The
        !            88: .UC .EQ
        !            89: and
        !            90: .UC .EN
        !            91: are copied through untouched;
        !            92: they
        !            93: are not otherwise processed
        !            94: by
        !            95: .UC EQN .
        !            96: This means that you have to take care
        !            97: of things like centering, numbering, and so on
        !            98: yourself.
        !            99: The most common way is to use the
        !           100: .UC TROFF
        !           101: and
        !           102: .UC NROFF
        !           103: macro package package `\(mims'
        !           104: developed by M. E. Lesk[3],
        !           105: which allows you to center, indent, left-justify and number equations.
        !           106: .PP
        !           107: With the `\(mims' package,
        !           108: equations are centered by default.
        !           109: To left-justify an equation, use
        !           110: .UC \&.EQ\ L
        !           111: instead of
        !           112: .UC .EQ .
        !           113: To indent it, use
        !           114: .UC .EQ\ I .
        !           115: Any of these can be followed by an arbitrary `equation number'
        !           116: which will be placed at the right margin.
        !           117: For example, the input
        !           118: .P1
        !           119: ^EQ I (3.1a)
        !           120: x = f(y/2) + y/2
        !           121: ^EN
        !           122: .P2
        !           123: produces the output
        !           124: .EQ I (3.1a)
        !           125: x = f(y/2) + y/2
        !           126: .EN
        !           127: .PP
        !           128: There is also a shorthand notation so
        !           129: in-line expressions
        !           130: like
        !           131: $pi sub i sup 2$
        !           132: can be entered without
        !           133: .UC .EQ
        !           134: and
        !           135: .UC .EN .
        !           136: We will talk about it in section 19.
        !           137: .SC Input spaces
        !           138: .PP
        !           139: Spaces and newlines within an expression are thrown away by
        !           140: .UC EQN .
        !           141: (Normal text is left absolutely alone.)
        !           142: Thus
        !           143: between
        !           144: .UC .EQ
        !           145: and
        !           146: .UC .EN ,
        !           147: .P1
        !           148: x=y+z
        !           149: .P2
        !           150: and
        !           151: .P1
        !           152: x = y + z
        !           153: .P2
        !           154: and
        !           155: .P1
        !           156: x   =   y   
        !           157:    + z
        !           158: .P2
        !           159: and so on
        !           160: all produce the same
        !           161: output
        !           162: .EQ
        !           163: x=y+z
        !           164: .EN
        !           165: You should use spaces and newlines freely to make your input equations
        !           166: readable and easy to edit.
        !           167: In particular, very long lines are a bad idea,
        !           168: since they are often hard to fix if you make a mistake.
        !           169: .SC Output spaces
        !           170: .PP
        !           171: To force extra spaces into the 
        !           172: .ul
        !           173: output,
        !           174: use a tilde ``\|~\|''
        !           175: for each space you want:
        !           176: .P1
        !           177: x~=~y~+~z
        !           178: .P2
        !           179: gives
        !           180: .EQ
        !           181: x~=~y~+~z
        !           182: .EN
        !           183: You can also use a circumflex ``^'', 
        !           184: which gives a space half the width of a tilde.
        !           185: It is mainly useful for fine-tuning.
        !           186: Tabs may also be used to position pieces
        !           187: of an expression,
        !           188: but the tab stops must be set by 
        !           189: .UC TROFF
        !           190: commands.
        !           191: .SC "Symbols, Special Names, Greek"
        !           192: .PP
        !           193: .UC EQN
        !           194: knows some mathematical symbols,
        !           195: some mathematical names, and the Greek alphabet.
        !           196: For example,
        !           197: .P1
        !           198: x=2 pi int sin ( omega t)dt
        !           199: .P2
        !           200: produces
        !           201: .EQ
        !           202: x = 2 pi int sin ( omega t)dt
        !           203: .EN
        !           204: Here the spaces in the input are
        !           205: .B
        !           206: necessary
        !           207: .R
        !           208: to tell
        !           209: .UC EQN
        !           210: that
        !           211: .ul
        !           212: int,
        !           213: .ul
        !           214: pi,
        !           215: .ul
        !           216: sin
        !           217: and
        !           218: .ul
        !           219: omega
        !           220: are separate entities that should get special treatment.
        !           221: The
        !           222: .ul
        !           223: sin,
        !           224: digit 2, and parentheses are set in roman type instead of italic;
        !           225: .ul
        !           226: pi
        !           227: and
        !           228: .ul
        !           229: omega
        !           230: are made Greek;
        !           231: and
        !           232: .ul
        !           233: int
        !           234: becomes the integral sign.
        !           235: .PP
        !           236: When in doubt, leave spaces around separate parts of the input.
        !           237: A
        !           238: .ul
        !           239: very
        !           240: common error is to type
        !           241: .ul
        !           242: f(pi)
        !           243: without leaving spaces on both sides of the
        !           244: .ul
        !           245: pi.
        !           246: As a result,
        !           247: .UC EQN
        !           248: does not recognize
        !           249: .ul
        !           250: pi
        !           251: as a special word, and it appears as
        !           252: $f(pi)$
        !           253: instead of
        !           254: $f( pi )$.
        !           255: .PP
        !           256: A complete list of
        !           257: .UC EQN
        !           258: names appears in section 23.
        !           259: Knowledgeable users can also use
        !           260: .UC TROFF
        !           261: four-character names
        !           262: for anything 
        !           263: .UC EQN
        !           264: doesn't know about,
        !           265: like
        !           266: .ul
        !           267: \\(bs
        !           268: for the Bell System sign \(bs.
        !           269: .SC "Spaces, Again"
        !           270: .PP
        !           271: The only way
        !           272: .UC EQN
        !           273: can deduce that some sequence
        !           274: of letters might be special
        !           275: is if that sequence is separated from the letters
        !           276: on either side of it.
        !           277: This can be done by surrounding a special word by ordinary spaces
        !           278: (or tabs or newlines),
        !           279: as we did in the previous section.
        !           280: .PP
        !           281: .tr ~~
        !           282: You can also make special words stand out by surrounding them
        !           283: with tildes or circumflexes:
        !           284: .P1
        !           285: x~=~2~pi~int~sin~(~omega~t~)~dt
        !           286: .P2
        !           287: is much the same as the last example,
        !           288: except that the tildes
        !           289: not only
        !           290: separate the magic words
        !           291: like
        !           292: .ul
        !           293: sin,
        !           294: .ul
        !           295: omega,
        !           296: and so on,
        !           297: but also add extra spaces,
        !           298: one space per tilde:
        !           299: .EQ
        !           300: x~=~2~pi~int~sin~(~omega~t~)~dt
        !           301: .EN
        !           302: .PP
        !           303: Special words can also be separated by braces { }
        !           304: and double quotes "...",
        !           305: which have special meanings that we will
        !           306: see soon.
        !           307: .tr ~
        !           308: .SC "Subscripts and Superscripts"
        !           309: .PP
        !           310: Subscripts and superscripts are
        !           311: obtained with the words
        !           312: .ul
        !           313: sub
        !           314: and
        !           315: .ul
        !           316: sup.
        !           317: .P1
        !           318: x sup 2 + y sub k
        !           319: .P2
        !           320: gives
        !           321: .EQ
        !           322: x sup 2 + y sub k
        !           323: .EN
        !           324: .UC EQN
        !           325: takes care of all the size changes and vertical motions
        !           326: needed to make the output look right.
        !           327: The words
        !           328: .ul
        !           329: sub
        !           330: and
        !           331: .ul
        !           332: sup
        !           333: must be surrounded by spaces;
        !           334: .ul
        !           335: x sub2
        !           336: will give you
        !           337: $x sub2$ instead of $x sub 2$.
        !           338: Furthermore, don't forget to leave a space
        !           339: (or a tilde, etc.)
        !           340: to mark the end of a subscript or superscript.
        !           341: A common error is to say
        !           342: something like
        !           343: .P1
        !           344: y = (x sup 2)+1
        !           345: .P2
        !           346: which causes
        !           347: .EQ
        !           348: y = (x sup 2)+1
        !           349: .EN
        !           350: instead of
        !           351: the intended
        !           352: .EQ
        !           353: y = (x sup 2 )+1
        !           354: .EN
        !           355: .PP
        !           356: Subscripted subscripts and superscripted superscripts
        !           357: also work:
        !           358: .P1
        !           359: x sub i sub 1
        !           360: .P2
        !           361: is
        !           362: .EQ
        !           363: x sub i sub 1
        !           364: .EN
        !           365: A subscript and superscript on the same thing
        !           366: are printed one above the other
        !           367: if the subscript comes
        !           368: .ul
        !           369: first:
        !           370: .P1
        !           371: x sub i sup 2
        !           372: .P2
        !           373: is
        !           374: .EQ
        !           375: x sub i sup 2
        !           376: .EN
        !           377: .PP
        !           378: Other than this special case,
        !           379: .ul
        !           380: sub
        !           381: and
        !           382: .ul
        !           383: sup
        !           384: group to the right, so
        !           385: .ul
        !           386: x\ sup\ y\ sub\ z
        !           387: means
        !           388: $x sup {y sub z}$, not ${x sup y} sub z$.
        !           389: .SC "Braces for Grouping"
        !           390: .PP
        !           391: Normally, the end of a subscript or superscript is marked
        !           392: simply by a blank (or tab or tilde, etc.)
        !           393: What if the subscript or superscript is something that has to be typed
        !           394: with blanks in it?
        !           395: In that case, you can use the braces
        !           396: { and } to mark the
        !           397: beginning and end of the subscript or superscript:
        !           398: .P1
        !           399: e sup {i omega t}
        !           400: .P2
        !           401: is
        !           402: .EQ
        !           403: e sup {i omega t}
        !           404: .EN
        !           405: .sp
        !           406: Rule:  Braces can
        !           407: .ul
        !           408: always
        !           409: be used to force 
        !           410: .UC EQN
        !           411: to treat something as a unit,
        !           412: or just to make your intent perfectly clear.
        !           413: Thus:
        !           414: .P1
        !           415: x sub {i sub 1} sup 2
        !           416: .P2
        !           417: is
        !           418: .EQ
        !           419: x sub {i sub 1} sup 2
        !           420: .EN
        !           421: with braces, but
        !           422: .P1
        !           423: x sub i sub 1 sup 2
        !           424: .P2
        !           425: is
        !           426: .EQ
        !           427: x sub i sub 1 sup 2
        !           428: .EN
        !           429: which is rather different.
        !           430: .PP
        !           431: Braces can occur within braces if necessary:
        !           432: .P1
        !           433: e sup {i pi sup {rho +1}}
        !           434: .P2
        !           435: is
        !           436: .EQ
        !           437: e sup {i pi sup {rho +1}}
        !           438: .EN
        !           439: The general rule is that anywhere you could use some single
        !           440: thing like
        !           441: .ul
        !           442: x,
        !           443: you can use an arbitrarily complicated thing if you enclose
        !           444: it in braces.
        !           445: .UC EQN
        !           446: will look after all the details of positioning it and making
        !           447: it the right size.
        !           448: .PP
        !           449: In all cases, make sure you have the
        !           450: right number of braces.
        !           451: Leaving one out or adding an extra will cause 
        !           452: .UC EQN
        !           453: to complain bitterly.
        !           454: .PP
        !           455: Occasionally you will have to
        !           456: print braces.
        !           457: To do this,
        !           458: enclose them in double quotes,
        !           459: like "{".
        !           460: Quoting is discussed in more detail in section 14.
        !           461: .SC Fractions
        !           462: .PP
        !           463: To make a fraction,
        !           464: use the word
        !           465: .ul
        !           466: over:
        !           467: .P1
        !           468: a+b over 2c =1
        !           469: .P2
        !           470: gives
        !           471: .EQ
        !           472: a+b over 2c =1
        !           473: .EN
        !           474: The line is made the right length and positioned automatically.
        !           475: Braces can be used to make clear what goes over what:
        !           476: .P1
        !           477: {alpha + beta} over {sin (x)}
        !           478: .P2
        !           479: is
        !           480: .EQ
        !           481: {alpha + beta} over {sin (x)}
        !           482: .EN
        !           483: What happens when there is both an
        !           484: .ul
        !           485: over
        !           486: and a
        !           487: .ul
        !           488: sup
        !           489: in the same expression?
        !           490: In such an apparently ambiguous case,
        !           491: .UC EQN
        !           492: does the
        !           493: .ul
        !           494: sup
        !           495: before the
        !           496: .ul
        !           497: over,
        !           498: so
        !           499: .P1
        !           500: \(mib sup 2 over pi
        !           501: .P2
        !           502: is
        !           503: $-b sup 2 over pi$
        !           504: instead of
        !           505: $-b sup {2 over pi}$
        !           506: The rules
        !           507: which decide which operation is done first in cases like this
        !           508: are summarized in section 23.
        !           509: When in doubt, however,
        !           510: .ul
        !           511: use braces
        !           512: to make clear what goes with what.
        !           513: .SC "Square Roots"
        !           514: .PP
        !           515: To draw a square root, use
        !           516: .ul
        !           517: sqrt:
        !           518: .P1 2
        !           519: sqrt a+b + 1 over sqrt {ax sup 2 +bx+c}
        !           520: .P2
        !           521: is
        !           522: .EQ
        !           523: sqrt a+b + 1 over sqrt {ax sup 2 +bx+c}
        !           524: .EN
        !           525: Warning _ square roots of tall quantities look lousy,
        !           526: because a root-sign 
        !           527: big enough to cover the quantity is
        !           528: too dark and heavy:
        !           529: .P1
        !           530: sqrt {a sup 2 over b sub 2}
        !           531: .P2
        !           532: is
        !           533: .EQ
        !           534: sqrt{a sup 2 over b sub 2}
        !           535: .EN
        !           536: Big square roots are generally better written as something
        !           537: to the power \(12:
        !           538: .EQ
        !           539: (a sup 2 /b sub 2 ) sup half
        !           540: .EN
        !           541: which is
        !           542: .P1
        !           543: (a sup 2 /b sub 2 ) sup half
        !           544: .P2
        !           545: .SC "Summation, Integral, Etc."
        !           546: .PP
        !           547: Summations, integrals, and similar constructions
        !           548: are easy:
        !           549: .P1
        !           550: sum from i=0 to {i= inf} x sup i
        !           551: .P2
        !           552: produces
        !           553: .EQ
        !           554: sum from i=0 to {i= inf} x sup i
        !           555: .EN
        !           556: Notice that we used
        !           557: braces to indicate where the upper
        !           558: part
        !           559: $i= inf$
        !           560: begins and ends.
        !           561: No braces were necessary for the lower part $i=0$,
        !           562: because it contained no blanks.
        !           563: The braces will never hurt,
        !           564: and if the 
        !           565: .ul
        !           566: from
        !           567: and
        !           568: .ul
        !           569: to
        !           570: parts contain any blanks, you must use braces around them.
        !           571: .PP
        !           572: The
        !           573: .ul
        !           574: from
        !           575: and
        !           576: .ul
        !           577: to
        !           578: parts are both optional,
        !           579: but if both are used,
        !           580: they have to occur in that order.
        !           581: .PP
        !           582: Other useful characters can replace the
        !           583: .ul
        !           584: sum
        !           585: in our example:
        !           586: .P1
        !           587: int   prod   union   inter
        !           588: .P2
        !           589: become, respectively,
        !           590: .EQ
        !           591: int ~~~~~~ prod ~~~~~~ union ~~~~~~ inter
        !           592: .EN
        !           593: Since the thing before the 
        !           594: .ul
        !           595: from
        !           596: can be anything,
        !           597: even something in braces,
        !           598: .ul
        !           599: from-to
        !           600: can often be used in unexpected ways:
        !           601: .P1
        !           602: lim from {n \(mi> inf} x sub n =0
        !           603: .P2
        !           604: is
        !           605: .EQ
        !           606: lim from {n-> inf} x sub n =0
        !           607: .EN

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