## How Mathematicians Discover Formulas in Differential Equations

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.
- (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.

- 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^{-1}*a*_{0} +
**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.

- 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].

- 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].