User:Ricardo sandoval

My main interest is in math articles.

Main contributions

Trigonometric functions: One paragraph near the end talking about the harmonic motion. I was impressed no one did that before, since it is a very important physics concept. Also pointed trigonometric functions are projections of the circular movement. And explained the animation on the side. Added table for tangents.

To do: The circular movement explains the derivatives of the sine and cosine very nicely, I wonder were that could fit.

Euler's formula: A demonstration of the Euler formula in the section differential equations proof (e^{ix}'=ie^{ix}). I think this demonstration is more direct and intuitive. Observation at the beginning of proofs, since problems with rigor as definition of the e^{ix} used is not cited in some proofs. Added a definition by limit of e^{z} as lim(1+z/n)^n. (not on right now its all under discussion)

To do: Don't know what to make of some of the other proofs.

Thales theorem(circle-triangle): Added a geometrical proof of the converse, I think it makes the converse more intuitive.

To do: Add a picture showing the rectangle and the half right triangle.(already done by someone else, Thank you!)

Minor changes

Golden ratio: Tried a clearer wording for the lead and calculation parts(done?)

Sine law: Added 2R = \frac{abc} {2A} to make the equation more understandable and useful.

Pythagorean theorem: Added the "If the angle between the sides is right it reduces to the Pythagorean theorem" to make the citation of the cosine law more understandable.

Heron's formula: Completed the steps on the demonstration using difference of squares.

Pi: Added that the ratio c/d is always the same so the definition makes sense.

Rectangle: Added that the diagonal crosses at he midpoints, are equal, and can be calculated using Pythagoras.

Complex numbers: Added a link to Visual complex Analysis a book by Tristan Needham.

Trapezium: Area formula based on its sides relates to Heron's formula and fails when parallel sides are equal(explaining why geometrically).

Quadratic function: Explained the effect of each term on the graph. Expanded the maximum/minimum analysis for the bivariate case.

Arithmetic Progression: Added a_n=a_m+(n-m)*r there a nice interpretation for that(add it somewhere?)

Exponentiation: Changed wording at the lead, some details on principal root and rational powers.

Brahmagupta-Fibonacci identity added relation to absolute property of complex numbers.

Derivative take out POV in the section generalization in "a 'very' important generalization" and "a 'natural' generalization".

Tangent changed second paragraph of calculus section (very rough)

Inverse trigonometric functions changed order of relationships, and excluded i and -i from the power series of arctan.

Congruence (geometry) details for the SSA ambiguity.

Cosine law same details as above.

To do:

Revise the Cosine law, already found that the power circle proofs can be simplified(the French one I think its too simplified)

Pythagoras proven equivalent to the parallel postulate?

Pyramid needs demonstrations.

Congruence add that every non-isosceles triangle has two different versions(reflected, non reflected)?

Brahmagupta-Fibonacci identity some simplification is needed?

Circle add section of circles on nature.

Pi correct the Indian power series entry.

Heron's formula give the geometrical demonstration.

Trigonometric identities give neat demonstration of tangent half angle formula?

Tangent make subsections, change second paragraph of calculus section.

Perceptions

Uniform circular motion article has many problems.

Complex numbers geometrical interpretation (as rotation and stretching) rules! but need to emphasize that more?

Determinant article is much better in French with nice geometric interpretations.

de Moivre's formula could be explained by multiple application of trig identities(not rigorous but more insightful).

Research in the literature
Most of the literature I saw on Euler's formula define first $$ e^z \,$$ for complex $$ z=x+yi \,$$ and definition (1) and (2) below are by far the most common.

(1) As the Taylor series:


 * $$ e^{z}= 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots \, $$

Curtiss(1978), Polya(1974), Courant(1965), Rudin(1966).

(2) I guess deceptively as


 * $$ e^{z}=e^x(\cos(y)+i\sin(y)) \, $$

Alhfors "Complex Analysis" (1953), Robert B. Ash "Complex Variables"(1971), Anthony B. Holland "Complex function Theory" (1980), Greene/ Krantz "Function Theory of One Complex Variable"(2002), T. Gamelin "Complex Analysis"(2001)

(3) As the limit:



e^{z}=\lim_{n\rarr\infty}\left(1+\frac{z}{n}\right)^n\, $$

E. Townsend "Functions of a complex variable" 1915, Feynman "Lectures on Physics" Algebra chapter.

(4) As the unique solution of the differential equation:


 * $$ f'(z)=f(z) \,$$ with $$ f(0)=1 \,$$

Lars V. Ahlfors "Complex analysis" (1966).

(5) By first defining $$ \log(z) \,$$

Hardy "Course of Pure mathematics" (1908)

Obs. By defining $$ \log(z) \,$$ as the line integral from 1 to z of 1/z it is needed to choose the "branch".

My opinions on Euler's formula
(in construction)

First I will add another definition

(6) Unique complex function $$ f \,$$ such that


 * $$ f(x+y)=f(x)f(y) \,$$ and $$ f'(0)=1 \,$$

Definition (2) avoids the problem of showing that such function exists by using Euler's formula.

All definitions above can be modified to get just $$ e^{ix} \, $$ for real $$ x \,$$.

Definitions (excepting (2)) can also be modified to get $$ e^x \,$$ for real $$ x \,$$.

One exercise would be to show they are all equivalent, and more so to prove directly the equivalence of each pair (15 of them). There would be also variations if you restricted to real, imaginary, or complex numbers.

I didn't do any research on definitions of $$ e^x \,$$ for real $$ x \,$$ but I would guess (1),(5) are the more common ones.

A modification of (6) is: the unique function $$ f(x)=a^x \,$$ such that $$ f'(0)=1 \,$$.

Conditions $$ f(x)=a^x \, $$ and $$ f(x+y)=f(x)f(y) \,$$ are not equivalent as one would expect and actually there are functions that satisfy the second that are discontinuous everywhere, I think this depends on the axiom of choice but I am not sure.

I think this is the best way to introduce the real exponential.

The many proofs
(in construction)

Lets divide the definition of exponential in cases

(A) Complex $$ z =x+yi$$

(B) Pure imaginary $$ yi$$

(C) Real $$ x $$

(3) implies (1)

using that

$$ (a+b)^n=\sum_{k=0}^n \frac{n!}{k!\,(n-k)!} a^{n-k}b^k \ $$

combined with

$$ e^x = \lim_{n \rightarrow \infty} \left(1 + \frac{x}{n} \right)^n \ $$

is

$$ e^x = \lim_{n \rightarrow \infty} \sum_{k=0}^n \frac{n!}{k!\,(n-k)!} \frac{x^k}{n^k} \ $$

as n gets large, the early terms of the summation (where k<<n) become

$$ e^x = \lim_{n \rightarrow \infty} \sum_{k=0}^n \frac{(n)(n-1)(n-2)...(n-(k-1))}{k!} \frac{x^k}{n^k} \ $$

$$ = \lim_{n \rightarrow \infty} \sum_{k=0}^n \frac{(n)(n-1)(n-2)...(n-(k-1))}{n^k} \frac{x^k}{k!} \ $$

$$ = \lim_{n \rightarrow \infty} \sum_{k=0}^n (1) \left(1-\frac{1}{n}\right) \left(1-\frac{2}{n}\right)...\left(1-\frac{k-1}{n}\right) \frac{x^k}{k!} \ $$

$$ \rightarrow \sum_{k=0}^{\infty} \frac{x^k}{k!} \ $$

In fact the finite sums at the right converge to the infinite sum so their diference can be made as smaal as one wants and each term on the initial sum converges to the corresponding term at the end so a finite sum of the terms at the initial sequence can be made as close as one wants to the partial sum at the right.

2B + (trig identities) implies 3B

you can prove

$$ (\cos(x)+i\sin(x)) (\cos(y)+i\sin(y)) = \cos(x+y)+i\sin(x+y) \ \,$$

by trigonometric identities, then

$$ (\cos(x)+i\sin(x))^n = \cos(nx)+i\sin(nx) \ \,$$

by induction or by multiple application of the last one.

$$ \cos(x)+i\sin(x)= (\cos(\frac{x}{n})+i\sin(\frac{x}{n}))^n \,$$

When n is big x/n is small so cos(x/n) is almost 1 and sin(x/n) almost x/n.

$$\approx (1+i\frac{x}{n})^n \,$$ for big n. So we should have

$$ \cos(x)+i\sin(x)= \lim_{n \rightarrow \infty} (1+i\frac{x}{n})^n \,$$

To make it fully rigorous its kind of painful.