Complex logarithm

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a complex logarithm is a generalization of the <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a> to nonzero <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>. The term refers to one of the following, which are strongly related:

<ul><li>A complex logarithm of a nonzero complex number 
 
 
 
 z
 
 
 {\displaystyle z}
 
, defined to be any complex number 
 
 
 
 w
 
 
 {\displaystyle w}
 
 for which 
 
 
 
 
 e
 
 w
 
 
 =
 z
 
 
 {\displaystyle e^{w}=z}
 
. Such a number 
 
 
 
 w
 
 
 {\displaystyle w}
 
 is denoted by 
 
 
 
 log
 ⁡
 z
 
 
 {\displaystyle \log z}
 
. If 
 
 
 
 z
 
 
 {\displaystyle z}
 
 is given in <a href="/facts/Polar_form/2w7mqI7e">polar form</a> as 
 
 
 
 z
 =
 r
 
 e
 
 i
 θ
 
 
 
 
 {\displaystyle z=re^{i\theta }}
 
, where 
 
 
 
 r
 
 
 {\displaystyle r}
 
 and 
 
 
 
 θ
 
 
 {\displaystyle \theta }
 
 are real numbers with 
 
 
 
 r
 >
 0
 
 
 {\displaystyle r>0}
 
, then 
 
 
 
 ln
 ⁡
 r
 +
 i
 θ
 
 
 {\displaystyle \ln r+i\theta }
 
 is one logarithm of 
 
 
 
 z
 
 
 {\displaystyle z}
 
, and all the complex logarithms of 
 
 
 
 z
 
 
 {\displaystyle z}
 
 are exactly the numbers of the form 
 
 
 
 ln
 ⁡
 r
 +
 i
 
 (
 
 θ
 +
 2
 π
 k
 
 )
 
 
 
 {\displaystyle \ln r+i\left(\theta +2\pi k\right)}
 
 for integers 
 
 
 
 k
 
 
 {\displaystyle k}
 
. These logarithms are equally spaced along a vertical line in the complex plane.</li>
<li>A complex-valued function 
 
 
 
 log
 :
 U
 →
 
 C
 
 
 
 {\displaystyle \log \colon U\to \mathbb {C} }
 
, defined on some subset 
 
 
 
 U
 
 
 {\displaystyle U}
 
 of the set 
 
 
 
 
 
 C
 
 
 ∗
 
 
 
 
 {\displaystyle \mathbb {C} ^{*}}
 
 of nonzero complex numbers, satisfying 
 
 
 
 
 e
 
 log
 ⁡
 z
 
 
 =
 z
 
 
 {\displaystyle e^{\log z}=z}
 
 for all 
 
 
 
 z
 
 
 {\displaystyle z}
 
 in 
 
 
 
 U
 
 
 {\displaystyle U}
 
. Such complex logarithm functions are analogous to the real <a href="/facts/Natural_logarithm/PnreKEzI">logarithm function</a> 
 
 
 
 ln
 :
 
 
 R
 
 
 >
 0
 
 
 →
 
 R
 
 
 
 {\displaystyle \ln \colon \mathbb {R} _{>0}\to \mathbb {R} }
 
, which is the <a href="/facts/Inverse_function/Tg5nnXlu">inverse </a> of the real <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a> and hence satisfies eln x = x for all positive real numbers x. Complex logarithm functions can be constructed by explicit formulas involving real-valued functions, by integration of 
 
 
 
 1
 
 /
 
 z
 
 
 {\displaystyle 1/z}
 
, or by the process of <a href="/facts/Analytic_continuation/aBCuEgvt">analytic continuation</a>.</li></ul>
There is no <a href="/facts/Continuous_function/sKbl02pB">continuous</a> complex logarithm function defined on all of 
 
 
 
 
 
 C
 
 
 ∗
 
 
 
 
 {\displaystyle \mathbb {C} ^{*}}
 
. Ways of dealing with this include <a href="/facts/Branch_point/lJ7Y6Iht">branches</a>, the associated <a href="/facts/Riemann_surface/qoj8ogv8">Riemann surface</a>, and <a href="/facts/Partial_inverse/Tg5nnXlu">partial inverses</a> of the <a href="/facts/Exponential_function/ewlYxvR2">complex exponential function</a>. The principal value defines a particular complex logarithm function 
 
 
 
 Log
 :
 
 
 C
 
 
 ∗
 
 
 →
 
 C
 
 
 
 {\displaystyle \operatorname {Log} \colon \mathbb {C} ^{*}\to \mathbb {C} }
 
 that is continuous except along the negative real axis; on the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> with the negative real numbers and 0 removed, it is the <a href="/facts/Analytic_continuation/aBCuEgvt">analytic continuation</a> of the (real) natural logarithm.

Complex logarithm open-in-new

Complex logarithm