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=== Logaritma kompleks ===
{{Main|Complex logarithm}}
[[Berkas:Complex_number_illustration_multiple_arguments.svg|al=AnSebuah illustrationilustrasi ofmengenai thebentuk polar form: asebuah pointtitik isyang describeddijelaskan bymelalui ansebuah arrowpanah oratau equivalentlysecara byekuivalen itsmelalui lengthpanjang anddan anglesudutnya to theke sumbu-x-axis.|ka|jmpl|PolarBentuk formpolar ofdari {{math|''z {{=}} x + iy''}}. Both {{mvar|φ}} anddan {{mvar|φ'}} aremerupakan argumentsargumen ofdari {{mvar|z}}.]]
All theSemua [[Complexbilangan number|complex numberskompleks]] {{mvar|a}} that solveyang themenyelesaikan equationpersamaan
: <math>e^a=z</math>
are calleddisebut ''complexlogaritma logarithmskompleks'' ofdari {{mvar|z}}, whenketika {{mvar|z}} is (considereddianggap assebagai) abilangan complex numberkompleks. A complex numberBilangan iskompleks commonlybiasanya representeddinyatakan assebagai {{math|''z {{=}} x + iy''}}, wheredengan {{mvar|x}} anddan {{mvar|y}} aremerupakan realbilangan numbersreal anddan {{mvar|i}} is anmerupakan [[imaginarysatuan unitimajiner]], the(satuan squareyang ofdikuadratkan whichmemberikan isnilai −1). Such a number canBilangan bekompleks visualizeddapat bydivisualisasikan amelalui pointsebuah intitik thedalam [[complexbidang planekompleks]], asseperti shownyang atdiperlihatkan thepada rightgambar berikut. The [[Bentuk polar form]] encodesmenulis abilangan nonkompleks tak-zero complex numbernol {{mvar|z}} bymelalui itstitik [[absolutenilai valuemutlak]], thatyang is,berarti thejarak (positive,yang berupa bilangan bernilai real) distancedan positif {{Mvar|r}} tosama thedengan titik {{mvar|z}} ke [[OriginTitik asal (mathematicsmatematika)|origintitik asalnya]],. andBentuk anpolar anglejuga betweenmenulis thesebuah sudut antara bilangan real pada sumbu-{{Math|Re}} (yakni sumbu-{{mvar|x}}) axis'' ''{{Math|Re}} anddan thegaris lineyang passingmelalui throughtitik bothasal thedan origin andtitik {{mvar|z}}. This angleSudut istersebut calleddisebut thesebagai [[ArgumentArgumen (complexbilangan analysiskompleks)|argumentargumen]] ofdari {{mvar|z}}.
TheNilai absolute valuemutlak {{mvar|r}} ofdari {{mvar|z}} is givendinyatakan bysebagai
: <math>\textstyle r=\sqrt{x^2+y^2}.</math>
UsingDengan themenggunakan geometricalpandangan interpretationgeometris offungsi [[sinesinus (matematika)|sinus]] anddan [[cosinekosinus (matematika)]] andbeserta theirperiodisitasnya periodicity indalam {{Math|2{{pi}}}}, anysetiap complexbilangan numberkompleks {{mvar|z}} may bedapat denoteddinyatakan assebagai
: <math>z = x + iy = r (\cos \varphi + i \sin \varphi )= r (\cos (\varphi + 2k\pi) + i \sin (\varphi + 2k\pi)),</math>
foruntuk anysetiap integerbilangan numberbulat {{mvar|k}}. Evidently the argument of {{mvar|z}} is not uniquely specified: both {{mvar|φ}} and {{Math|1=''φ''' = ''φ'' + 2''k''{{pi}}}} are valid arguments of {{mvar|z}} for all integers {{mvar|k}}, because adding {{Math|2''k''{{pi}}}} [[Radian|radians]] or ''k''⋅360°{{refn|See [[radian]] for the conversion between 2[[pi|{{pi}}]] and 360 [[degree (angle)|degree]].|group=nb}} to {{mvar|φ}} corresponds to "winding" around the origin counter-clock-wise by {{mvar|k}} [[Turn (geometry)|turns]]. The resulting complex number is always {{mvar|z}}, as illustrated at the right for {{math|''k'' {{=}} 1}}. One may select exactly one of the possible arguments of {{mvar|z}} as the so-called ''principal argument'', denoted {{math|Arg(''z'')}}, with a capital {{math|A}}, by requiring {{mvar|φ}} to belong to one, conveniently selected turn, e.g. {{Math|−{{pi}} < ''φ'' ≤ {{pi}}}}<ref>{{Citation|last1=Ganguly|location=Kolkata|first1=S.|title=Elements of Complex Analysis|publisher=Academic Publishers|isbn=978-81-87504-86-3|year=2005}}, Definition 1.6.3</ref> or {{Math|0 ≤ ''φ'' < 2{{pi}}}}.<ref>{{Citation|last1=Nevanlinna|first1=Rolf Herman|author1-link=Rolf Nevanlinna|last2=Paatero|first2=Veikko|title=Introduction to complex analysis|journal=London: Hilger|location=Providence, RI|publisher=AMS Bookstore|isbn=978-0-8218-4399-4|year=2007|bibcode=1974aitc.book.....W}}, section 5.9</ref> These regions, where the argument of {{mvar|z}} is uniquely determined are called [[Principal branch|''branches'']] of the argument function.
[[Berkas:Complex_log_domain.svg|al=A density plot. In the middle there is a black point, at the negative axis the hue jumps sharply and evolves smoothly otherwise.|ka|jmpl|The principal branch (-{{pi}}, {{pi}}) of the complex logarithm, {{math|Log(''z'')}}. The black point at {{math|''z'' {{=}} 1}} corresponds to absolute value zero and brighter colors refer to bigger absolute values. The [[hue]] of the color encodes the argument of {{math|Log(''z'')}}.]]
[[Euler's formula]] connects the [[trigonometric functions]] [[sine]] and [[cosine]] to the [[complex exponential]]:
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