Logarithm (multivalued)

The (multivalued) logarithm function $\mathrm{Log} \colon \mathbb{C} \rightarrow \mathscr{P}\left( \mathbb{C} \right)$ is defined by $$\mathrm{Log}(z)=\displaystyle\int_1^z \dfrac{1}{t} \mathrm{d}t,$$ where $\mathscr{P} \left( \mathbb{C} \right)$ denotes the power set of $\mathbb{C}$ and where we understand the integral $\displaystyle\int_1^z$ as a contour integral over a path from $1$ to $z$ that does not cross $0$.

Properties

Real and imaginary parts of log
Logarithm (multivalued) of product is a sum of logarithms (multivalued)
Logarithm (multivalued) of a quotient is a difference of logarithms (multivalued)
Relationship between logarithm (multivalued) and positive integer exponents

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): 4.1.4