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As Cauchy was the founder of complex analysis (see "The origins of complex analysis"), a large proportion of the results now recognized as fundamental in this field bear his name today. One notable exception is Liouville's theorem, which states that a holomorphic function (that is, one differentiable in the complex sense) is bounded only if it is constant. Without going into detail, this theorem provides valuable information about the constraints governing a vast class of functions.
Cauchy claimed credit for this result, but to no avail. Admittedly, it can be obtained as a relatively straightforward consequence of his own research. If posterity did not grant his claim, it is because Cauchy seems quite simply never to have thought of stating the result. Credit therefore goes to the mathematician Joseph Liouville (1809–1882)—even though there appears to be no explicit trace of it in his work either!
Cauchy? No: Poisson!
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Let C be a circle with center O and radius 1, and let line D be tangent to the circle at a point A. Choose a point P at random on the circle, and mark the point Q where line (OP ) intersects line D. In probabilistic terms, the signed length AQ (taken as positive on one side of A and negative on the other) has a Cauchy distribution (see "A distribution unlike any other"). This probability distribution has the unexpected property of having no mean: if the experiment is repeated and the average of the successive lengths AQ is calculated, it will generally not approach a limiting value—not even 0, despite the experiment's symmetry.
