Mathematicians may discover a formula using a loose but quick argument. Their techniques may help us understand the insight and the essence of the formulas, and may be used as the first step toward building rigorous proofs. Their approach is useful in predicting the form of the formula. Their discovery is important in a theory's development.

1. (The inversion theorem) Rudin's diagram given in [Ru3, p.xii] says that in order to understand the inversion theorem [Ru3, p.170, Theorem 7.7 (2)], one should read at least the materials in [Ru3, chaps. 1,2,3,6; pp.166-170]. Even if one is lucky enough to overcome all the obstacles along the way and understand every step of the proof of [Ru3, Theorem 7.7], the insight and the essence of the theorem may still evade one's grips. Therefore, it is important to learn how mathematicians use keen observations to discover formulas. It is amazing that Cauchy derived  this formula in a few lines [Tit, p.3, l.5-l.11] even though his argument is somewhat loose.



2. Fourier's integral formula: From Fourier series to Fourier integrals
       Fourier formally derived Fourier's integral formula [Tit, p.1, (1.1.1)] in two steps [Tit, p.1, l.6-l.17]:
   Step I. Use the following corresponding formula in Fourier series:
   f (x) = 2^-1a₀ + S^¥_n=1 (a_n cos nx + b_n sin nx).
   Change x to x/l and then associate l with n.
   Step II. In order to expand the domain of n from integers to real numbers, we must divide n by l, where l is large. Then let the quotient be a real variable u. We have du = 1/l.
   Remark. One may compare Fourier's argument with Zygmund's proof of [Zyg, vol.2, p.246, Theorem 1.21]. I read [Zyg] years ago. Several months ago I tried to review this proof of Zygmund's. I found I could not understand the proof by reading [Zyg, vol.2, chap.XVI] alone. Only after I viewed [Zyg, vol.1, chaps. I, II, and III] did the proof of [Zyg, vol.2, p.246, Theorem 1.21] start to make sense to me.



3. We may naturally and rigorously derive Fourier's formula for series given in [Tit, p.6, l.7] from Fourier's integral formula [Tit, p.3, (1.1.6) or p.5, (1.3.4)] by using the calculus of residues.
   Proof. [Tit, p.5, l.-6-p.6, l.8].



4. A bumper crop often comes from a good observation.
   Example: Volterra observed that the integral equation of the second kind could be regarded as a limiting form of a system of linear equations  [Wat1, p.213, l.14-l.16]. Consequently, their solving methods are similar and similarly delicate [Boc, §7 & §8].