section_path: Generalizations/Complex logarithm
section_name: Complex logarithm
paragraph_text: Complex logarithm
  
  All the complex numbers a that solve the equation
  
  e a = z {\displaystyle e^{a}=z}
  
  are called complex logarithms of z, when z is (considered as) a complex number. A complex number is commonly represented as z = x + iy, where x and y are real numbers and 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 z by its absolute value, that is, the (positive, real) distance r to the origin, and an angle between the real (x) axis Re and the line passing through both the origin and z. This angle is called the argument of z.
  
  The absolute value r of z is given by
  
  r = x 2 + y 2. {\displaystyle \textstyle r={\sqrt {x^{2}+y^{2}}}.}
  
  Using the geometrical interpretation of sine and cosine and their periodicity in 2 π, any complex number z may be denoted as
  
  z = x + i y = r (cos φ + i sin φ) = r (cos (φ + 2 k π) + i sin (φ + 2 k π)), {\displaystyle z=x+iy=r(\cos \varphi +i\sin \varphi)=r(\cos(\varphi +2k\pi)+i\sin(\varphi +2k\pi)),}
  
  for any integer number k. Evidently the argument of z is not uniquely specified: both φ and φ' = φ + 2 k π are valid arguments of z for all integers k, because adding 2 k π radians or k ⋅360° to φ corresponds to "winding" around the origin counter-clock-wise by k turns. The resulting complex number is always z, as illustrated at the right for k = 1. One may select exactly one of the possible arguments of z as the so-called principal argument, denoted Arg(z), with a capital A, by requiring φ to belong to one, conveniently selected turn, e.g. − π < φ ≤ π or 0 ≤ φ < 2 π. These regions, where the argument of z is uniquely determined are called branches of the argument function.
  
  Euler's formula connects the trigonometric functions sine and cosine to the complex exponential: e i φ = cos φ + i sin φ. {\displaystyle e^{i\varphi }=\cos \varphi +i\sin \varphi.}
  
  Using this formula, and again the periodicity, the following identities hold:
  
  z = r (cos φ + i sin φ) = r (cos (φ + 2 k π) + i sin (φ + 2 k π)) = r e i (φ + 2 k π) = e ln (r) e i (φ + 2 k π) = e ln (r) + i (φ + 2 k π) = e a k, {\displaystyle {\begin{array}{lll}z&=&r\left(\cos \varphi +i\sin \varphi \right)\\&=&r\left(\cos(\varphi +2k\pi)+i\sin(\varphi +2k\pi)\right)\\&=&re^{i(\varphi +2k\pi)}\\&=&e^{\ln(r)}e^{i(\varphi +2k\pi)}\\&=&e^{\ln(r)+i(\varphi +2k\pi)}=e^{a_{k}},\end{array}}}
  
  where ln(r) is the unique real natural logarithm, a denote the complex logarithms of z, and k is an arbitrary integer. Therefore, the complex logarithms of z, which are all those complex values a for which the a -th power of e equals z, are the infinitely many values
  
  Taking k such that φ + 2 k π is within the defined interval for the principal arguments, then a is called the principal value of the logarithm, denoted Log(z), again with a capital L. The principal argument of any positive real number x is 0; hence Log(x) is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers do not generalize to the principal value of the complex logarithm.
  
  The illustration at the right depicts Log(z), confining the arguments of z to the interval (−π, π ]. This way the corresponding branch of the complex logarithm has discontinuities all along the negative real x 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 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 z ", and consequently the "logarithm of z ", multi-valued functions.