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The standard branch of the matrix logarithm

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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:

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