Editing Logarithm (section)

===Complex logarithm===
{{Main|Complex logarithm}}
[[File:Complex number illustration multiple arguments.svg|thumb|right|Polar form of {{math|''z {{=}} x + iy''}}. Both {{mvar|φ}} and {{mvar|φ'}} are arguments of {{mvar|z}}.|alt=An illustration of the polar form: a point is described by an arrow or equivalently by its length and angle to the x-axis.]]

All the [[complex number]]s {{mvar|a}} that solve the equation

<math display="block">e^a=z</math>

are called ''complex logarithms'' of {{mvar|z}}, when {{mvar|z}} is (considered as) a complex number. A complex number is commonly represented as {{math|''z {{=}} x + iy''}}, where {{mvar|x}} and {{mvar|y}} are real numbers and {{mvar|i}} is an [[imaginary unit]], the square of which is −1. Such a number can be visualized by a point in the [[complex plane]], as shown at the right. The [[polar form]] encodes a non-zero complex number&nbsp;{{mvar|z}} by its [[absolute value]], that is, the (positive, real) distance&nbsp;{{Mvar|r}} to the [[origin (mathematics)|origin]], and an angle between the real ({{mvar|x}}) axis''&nbsp;''{{Math|Re}} and the line passing through both the origin and {{mvar|z}}. This angle is called the [[Argument (complex analysis)|argument]] of {{mvar|z}}.

The absolute value {{mvar|r}} of {{mvar|z}} is given by

<math display="block">\textstyle r=\sqrt{x^2+y^2}.</math>

Using the geometrical interpretation of [[sine]] and [[cosine]] and their periodicity in {{Math|2{{pi}}}}, any complex number&nbsp;{{mvar|z}} may be denoted as

<math display="block">\begin{align}
z &= x + iy \\
  &= r (\cos \varphi + i \sin \varphi ) \\
  &= r (\cos (\varphi + 2k\pi) + i \sin (\varphi + 2k\pi)),
\end{align}</math>

for any integer number&nbsp;{{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&nbsp;{{mvar|k}}, because adding {{Math|2''k''{{pi}}}}&nbsp;[[radian]]s 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}}&nbsp;[[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&nbsp;{{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.

[[File:Complex log domain.svg|right|thumb|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'')}}.|alt=A density plot. In the middle there is a black point, at the negative axis the hue jumps sharply and evolves smoothly otherwise.]]

[[Euler's formula]] connects the [[trigonometric functions]] [[sine]] and [[cosine]] to the [[complex exponential]]:
<math display="block">e^{i\varphi} = \cos \varphi + i\sin \varphi .</math>

Using this formula, and again the periodicity, the following identities hold:<ref>{{Citation|last1=Moore|first1=Theral Orvis|last2=Hadlock|first2=Edwin H.|title=Complex analysis|publisher=[[World Scientific]]|location=Singapore|isbn=978-981-02-0246-0|year=1991}}, section 1.2</ref>

<math display="block"> \begin{align}
z &= r \left (\cos \varphi + i \sin \varphi\right) \\
  &= r \left (\cos(\varphi + 2k\pi) + i \sin(\varphi + 2k\pi)\right) \\
  &= r e^{i (\varphi + 2k\pi)} \\
  &= e^{\ln(r)} e^{i (\varphi + 2k\pi)}  \\
  &= e^{\ln(r) + i(\varphi + 2k\pi)} = e^{a_k},
\end{align}
</math>

where {{math|ln(''r'')}} is the unique real natural logarithm, {{math|''a''<sub>''k''</sub>}} denote the complex logarithms of {{mvar|z}}, and {{mvar|k}} is an arbitrary integer. Therefore, the complex logarithms of {{mvar|z}}, which are all those complex values {{math|''a''<sub>''k''</sub>}} for which the {{math|''a''<sub>''k''</sub>-th}}&nbsp;power of {{mvar|e}} equals {{mvar|z}}, are the infinitely many values
<math display="block">a_k = \ln (r) + i ( \varphi + 2 k \pi ),</math>
for arbitrary integers {{mvar|k}}.

Taking {{mvar|k}} such that {{Math|''φ'' + 2''k''{{pi}}}} is within the defined interval for the principal arguments, then {{math|''a''<sub>''k''</sub>}} is called the ''principal value'' of the logarithm, denoted {{math|Log(''z'')}}, again with a capital&nbsp;{{math|L}}. The principal argument of any positive real number&nbsp;{{mvar|x}} is 0; hence {{math|Log(''x'')}} is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers [[Exponentiation#Failure of power and logarithm identities|do {{em|not}} generalize]] to the principal value of the complex logarithm.<ref>{{Citation | last1=Wilde | first1=Ivan Francis | title=Lecture notes on complex analysis | publisher=Imperial College Press | location=London | isbn=978-1-86094-642-4 | year=2006|url=https://books.google.com/books?id=vrWES2W6vG0C&q=complex+logarithm&pg=PA97}}, theorem 6.1.</ref>

The illustration at the right depicts {{math|Log(''z'')}}, confining the arguments of {{mvar|z}} to the interval {{open-closed|−π, π}}. This way the corresponding branch of the complex logarithm has discontinuities all along the negative real {{mvar|x}}&nbsp;axis, which can be seen in the jump in the hue there. This discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e. not changing to the corresponding {{mvar|k}}-value of the continuously neighboring branch. Such a locus is called a [[branch cut]]. Dropping the range restrictions on the argument makes the relations "argument of {{mvar|z}}", and consequently the "logarithm of {{mvar|z}}", [[multi-valued function]]s.