Logarithm
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.
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