In 1862 Gustav Bauer married Amalie, daughter of the Archivrat and Professor Honorarius Nathanael von Schlichtegroll.
In 1857 Bauer spent three months in England and upon his return to Germany became a Privatdozent for the Mathematics Faculty of the Ludwig Maximilian University of Munich.
Bauer's mathematical research dealt with algebra, geometric problems, spherical harmonics, the gamma function, and generalized continued fractions.
His doctoral students include Heinrich Burkhardt, Eduard Ritter von Weber, and Christian August Vogler.
[5][6] Using a result of Bauer on generalized continued fractions, Oskar Perron published in 1952 the first proof of another formula of Ramanujan.