Definition:Complex Natural Logarithm/Historical Note

Historical Note on Complex Natural Logarithm

In $1702$, Johann Bernoulli encountered solutions of the primitive $\ds \int \dfrac {\d x} {a x^2 + b x + c}$ which seemed to require logarithms of complex numbers, which at that time had not been considered.

Johann Bernoulli and Gottfried Wilhelm von Leibniz both investigated, and by $1712$ they had developed opposing viewpoints on how to handle the logarithm of a negative number.

Bernoulli used the argument:

$\dfrac {\map \d {-x} } {-x} = \dfrac {\map \d x} x$, so by integration $\map \ln {-x} = \map \ln x$

while Leibniz insisted that the integration was only valid for positive $x$.

Leonhard Paul Euler noticed that the integration in question required a constant of integration, and so:

$\map \ln {-x} = \map \ln x + c$

where $c$ was necessarily imaginary.

This resolved the matter.

Sources

• 1983: Ian Stewart and David Tall: Complex Analysis (The Hitchhiker's Guide to the Plane) ... (previous) ... (next): $0$ The origins of complex analysis, and a modern viewpoint: $1$. The origins of complex numbers