Example
Problem. Find an upper bound for
where is the upper half-circle with radius traversed once in the counterclockwise direction.
Solution. First observe that the length of the path of integration is half the circumference of a circle with radius a, hence
Next we seek an upper bound M for the integrand when . By the triangle inequality we see that
therefore
because on . Hence
Therefore we apply the estimation lemma with M = 1 / (a2 − 1)2. The resulting bound is
Read more about this topic: Estimation Lemma
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“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)