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Jun
25
comment Solving Kepler's Equation
I don't know anything about your specific question, but in case you don't know of it, Peter Colwell published a historical survey book on Kepler's equation in 1993 -- Solving Kepler's Equation over Three Centuries.
Jun
25
comment Differential Equations: Confocal Ellipse and Hyperbola
Try a search for the phrases "differential equation" and "confocal ellipses" (searched together) in google-books, with the date setting "19th century". In about a minute I found the following two references that appear to discuss your question: p. 247 of Volume I of George Boole's A Treatise on Differential Equations and Glaisher's short note on pp. 46-47 in Volume 9 of the journal Messenger of Mathematics.
Jun
24
comment Is it feasible for a sophomore in high school (15 years old) to learn complex analysis?
Incidentally, I know of three older books similar in level to Churchill's and Silverman's books that are freely available on the internet: Burkhardt's book (1913 translation by Rasor) Theory of Functions of a Complex Variable, Pierpont's 1914 book Functions of a Complex Variable, and Harkness/Morley's 1898 book Introduction to the Theory Analytic Functions.
Jun
24
comment Is it feasible for a sophomore in high school (15 years old) to learn complex analysis?
If the advanced undergraduate level complex variables/analysis books wind up being too advanced for you at present, you might want to look at the books I cite in my answer to What are the “real math” connections between Euclidean Geometry and Complex Numbers?. Also, the first chapter or two of most of the standard texts at the level of Churchill's book should be understandable (elementary functions and CR-equations, the stuff before line integrals appear).
Jun
22
comment Generalization to higher dimensions of a statement about plane triangles
I don't know anything about this topic, but if you can read French it's probably worth looking at the French references I gave in my answer to Higher-dimensional Extension of Triangle Geometry?.
Jun
18
comment Need resources for difficult Algebraic Identites.
Also, several old algebra texts from the 1800s have these kinds of identities. For example, see see the books by George Chrystal, Elias Loomis, Charles Smith, William Steadman Aldis, and Isaac Todhunter that I posed links to in this 5 November 2009 sci.math post at Math Forum. All of these books are freely available on the internet.
Jun
18
comment Need resources for difficult Algebraic Identites.
Many math competition problem books have chapters containing what you're looking for, such as: Xu Jiagu, Lecture Notes on Mathematical Olympiad Courses [for Junior Section Volume 1], Mathematical Olympiad Series #6, World Scientific Publishing Company, 2010, xii + 170 pages. [See Lecture 6: Some Methods of Factorization on pp. 35-40]. At this time there appears to be a .pdf copy on the internet, but since this might not be legally available, I am not giving the URL here.
Jun
17
comment what is the geometric meaning of total variation for function with two variables?
It's old, but On definitions of bounded variation for functions of two variables by Clarkson/Adams [Trans. Amer. Math. Soc. 35 (1933), 824-854] might be of use as a survey of early work. Also, you might find more recent work by looking for papers that cite this one. (I don't know enough about this topic to directly address your question.)
Jun
17
comment Comparing two large numbers
Taking "compare" to be a more general term than you probably intended, I notice that $5^{44}$ has a units digit of $5$ and $4^{53}$ doesn't have a units digit of $5.$
Jun
15
comment What is a good book to learn all of precalculus?
You might want to look at Mary P. Dolciani's Modern Introductory Analysis. See the comments here and here. This was the standard U.S. high school "precalculus" text from the mid 1960s through the 1970s, but there are probably very few places it could be used now because such a course (in high school) is no longer confined to the upper 10% or so of the population.
Jun
13
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Jun
9
comment High school algebra textbooks for gifted students
Maybe Spivak isn't a good analogy for what I'm thinking of, because once you get above the upper 2% realm, my guess is that such students are better served by self-study of the many existing supplementary books such as the New Mathematical Library books from the early 1960s and the current books targeted to those interested in math competitions (maybe some of these could be of use to you). Moreover, I would be less inclined to drill heavily into formalism and logic and proof, and more inclined with giving neat ideas and techniques.
Jun
9
comment Why this function is Riemann integrable?
Fermé somme means this: For each $t \in [a,b]$ we have $\lim_{x \rightarrow t}f(x) = 0,$ where $x$ is restricted to belong to the interval $[a,b]$ when the limit $x \rightarrow t$ is taken.
Jun
9
comment High school algebra textbooks for gifted students
It occurred to me that some comments I made a few years ago might be of interest to anyone drawn to this question. Down a ways in this 6 April 2010 math-teach post at Math Forum I discuss my interest, in the late 1980s, about writing such a book and how it never happened. However, I have a huge amount of material collected for such a task and perhaps one day (e.g. after I'm able to retire from my "day job" and can devote the needed time to something like this) I might try doing it.
Jun
6
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Jun
5
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Jun
2
comment How do I figure out my math aspirations?
(continuation) About 7 years later (he did a lot of graduate work as an undergraduate), he finished with a Ph.D. in math from Rice University. He told me once that he credits that book as probably the single strongest influence in his decision to continue studying math.
Jun
2
comment How do I figure out my math aspirations?
(continuation) Given his background (multivariable calculus, cookbook differential equations, elementary linear algebra, some self-taught number theory and basic abstract algebra), I thought the Courant/Robbins book would be a perfect overview for him. It was. He told me the following fall semester that this book really turned him on to math (he was mostly interested in history until his last year of high school). (comment continues)
Jun
2
comment How do I figure out my math aspirations?
Consider spending the summer carefully reading through Courant/Robbins' What is Mathematics?. Others have praised this book for many years in internet posts and naturally I've cited the book many times, but I don't believe I've shared the following story online. In May 1998 a fairly strong high school student that I had taught several classes to during the previous two years graduated, and he asked me what I suggest he do with his upcoming summer vacation before beginning college. (comment continues)
Jun
2
comment Exponential and Hyperbolic Functions Before Power Series
You might want to look at the first two or three chapters in Hyperbolic Functions: with Configuration Theorems and Equivalent and Equidecomposable Figures, which is an English translation of essays originally published in Russian.