Editing Imaginary unit (section)

=== Exponential and logarithm ===

The [[complex exponential]] function relates complex addition in the domain to complex multiplication in the codomain. Real values in the domain represent scaling in the codomain (multiplication by a real scalar) with {{math|1}} representing multiplication by {{mvar|e}}, while imaginary values in the domain represent rotation in the codomain (multiplication by a unit complex number) with {{mvar|i}} representing a rotation by {{math|1}} radian. The complex exponential is thus a periodic function in the imaginary direction, with period {{math|2''πi''}} and image {{math|1}} at points {{math|2''kπi''}} for all integers {{mvar|k}}, a real multiple of the lattice of imaginary integers.

The complex exponential can be broken into [[even and odd functions|even and odd]] components, the [[hyperbolic functions]] {{math|cosh}} and {{math|sinh}} or the [[trigonometric functions]] {{math|cos}} and {{math|sin}}:

<math display=block>\exp z = \cosh z + \sinh z = \cos(-iz) + i\sin(-iz)</math>

[[Euler's formula]] decomposes the exponential of an imaginary number representing a rotation:

<math display="block">\exp i\varphi = \cos \varphi + i\sin \varphi.</math>

This fact can be used to demonstrate, among other things, the apparently counterintuitive result that <math>i^i</math> is a real number.<ref>{{Cite web |title=i to the i is a Real Number – Math Fun Facts |url=https://math.hmc.edu/funfacts/i-to-the-i-is-a-real-number/ |access-date=2024-08-22 |website=math.hmc.edu |language=en-US}}</ref>

The quotient {{math|1=coth ''z'' = cosh ''z'' / sinh ''z'',}} with appropriate scaling, can be represented as an infinite [[partial fraction decomposition]] as the sum of [[reciprocal function]]s translated by imaginary integers:<ref>Euler expressed the partial fraction decomposition of the trigonometric cotangent as <math display="inline">\pi \cot \pi z = \frac1z + \frac1{z-1} + \frac1{z+1} + \frac1{z-2} + \frac1{z+2} + \cdots .</math> {{pb}} {{cite journal |last=Varadarajan |first=V. S. |title=Euler and his Work on Infinite Series |journal=Bulletin of the American Mathematical Society |series=New Series |volume=44 |number=4 |year=2007 |pages=515–539 |doi=10.1090/S0273-0979-07-01175-5 |doi-access=free }}</ref>
<math display="block">
\pi \coth \pi z = \lim_{n\to\infty}\sum_{k=-n}^n \frac{1}{z + ki}.
</math>

Other functions based on the complex exponential are well-defined with imaginary inputs. For example, a number raised to the {{mvar|ni}} power is:
<math display="block">x^{n i} = \cos(n\ln x) + i \sin(n\ln x ).</math> 

Because the exponential is periodic, its inverse the [[complex logarithm]] is a [[multi-valued function]], with each complex number in the domain corresponding to multiple values in the codomain, separated from each-other by any integer multiple of {{math|2''πi''.}} One way of obtaining a single-valued function is to treat the codomain as a [[cylinder]], with complex values separated by any integer multiple of {{math|2''πi''}} treated as the same value; another is to take the domain to be a [[Riemann surface]] consisting of multiple copies of the complex plane stitched together along the negative real axis as a [[branch cut]], with each branch in the domain corresponding to one infinite strip in the codomain.<ref>{{Cite book |last=Gbur |first=Greg |author-link=Greg Gbur |year=2011 |title=Mathematical Methods for Optical Physics and Engineering |publisher=Cambridge University Press |url=https://www.worldcat.org/oclc/704518582 |isbn=978-0-511-91510-9 |pages=278–284|oclc=704518582 }}</ref> Functions depending on the complex logarithm therefore depend on careful choice of branch to define and evaluate clearly.

For example, if one chooses any branch where <math>\ln i = \tfrac12 \pi i</math> then when {{mvar|x}} is a positive real number,
<math display=block> \log_i x = -\frac{2i \ln x }{\pi}.</math>