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# The standard branch of the matrix logarithm

Because of Euler’s identity \$latex {e^{pi i} + 1 = 0}&fg=000000\$, the complex exponential is not injective: \$latex {e^{z + 2pi i k} = e^z}&fg=000000\$ for any complex \$latex {z}&fg=000000\$ and integer \$latex {k}&fg=000000\$. As such, the complex logarithm \$latex {z mapsto log z}&fg=000000\$ is not well-defined as a single-valued function from \$latex {{bf C} backslash {0}}&fg=000000\$ to \$latex {{bf C}}&fg=000000\$. However, after making a branch cut, one can create a branch of the logarithm which is single-valued. For instance, after removing the negative real axis \$latex {(-infty,0]}&fg=000000\$, one has the standard branch \$latex {hbox{Log}: {bf C} backslash (-infty,0] rightarrow { z in {bf C}: |hbox{Im} z| < pi }}&fg=000000\$ of the logarithm, with \$latex {hbox{Log}(z)}&fg=000000\$ defined as the unique choice of the complex logarithm of \$latex {z}&fg=000000\$ whose imaginary part has magnitude strictly less than \$latex {pi}&fg=000000\$. This particular branch has a number of useful additional properties:

• The…

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