lurflurf

It is best to not get too caught up in what different results are called. The various terms have connotations, but there are exceptions and inconsistencies. Don't get too caught up in whether something is a law, proposition, lemma, collorary, theorem, definition, or fact. Like they say "One womans theorem is anothers definition."

hint erf is an antiderivative of a gaussian what if the fourier transform of a gaussian?

See here http://mathworld.wolfram.com/Base.html 5.25 (base 10) is indeed 101.01 (base 2) The idea is to write x in base b use [math] x = \sum_{k = -\infty}^{n} a_n b^n [/math] where a(n) is an integer and 0<=a(n)<b and 1<b

In introductory Algebra one is usually concerned with algebraic structures and relationships between them. In previous work you have seen specific examples of groups, fields, rings, vector spaces, modules, division algebras, lie algebrasand others. Now you can learn about these stuctures in general, learn new specific examples, and how different stuctures relate. For example since log(x*y)=log(x)+log(y) for real numbers x,y>0 we see that in some sense multiplication of positive reals is doing the same thing as addition of reals. Group theory is a part of algebra, but it is incorrect that it is not used in applications. Group theory is used alot in chemisty, physics and enginearing. Though the perspective is different, they are less interested in general groups and are more interested in groups thar repersent symetry and the matrix repersentations of such groups. In calculus many new results are given and known results are proven. In many intro calculus books there are lots of phrases like "for proof consult a more advanced book" or "this step is valid but consult a more advanced book to see why". Those type of questions and more are answered. Also more general integrals are studied such as the Lebesgue Integral, Lebesgue-Stieltjes Integral, and Riemann-Stieltjes Integral. Of great interest is what happens when the order of certain limiting processes are interchanged. For example conditions under which a sum can be integrated term by term. The epsilon-delta limit proofs are used more than in most intro courses. Differential forms are often introduced as a new way to view vector calculus. Special funtions are studies such as the gamma function, hypergeometric functions, and functions arising as integrals of certain elementary functions such. There is often some topology which is the study of properties of sets that are invarient under continuous deformations. Sometimes there is some complex variable theory where calculus is done using complex numbers. In general calculus is considered from a more theoretical perspective and less time is spent finding integrals and derivatives of elementary functions.

Lots of interesting subjects are availible to studens with the equivelant of one year of college math and you have had two! You should try and decide where you interest lie. A good place to start given your requirements is one of the many introduction to applied math books (they are often titles something like advanced math for physics and enginearing). Such Books normally require a year of calculus, they include material on vector calculus, odes, and linear algebra (some of which you can skip if you know it); next they deal with some finite calculus, numerical methods, pdes, fourier series, integral transforms, complex variables, and calculus of variations. These books have a variety of interesting material and if one of their areas especially apeals to you you can find a nice book that deals with one of the areas in geater depth. Vectors, Tensors and the Basic Equations of Fluid Mechanics (Dover Books on Engineering) by Rutherford Aris is a nice book that assumes a year of calculus (if tensor or fluids interest you). Also you could start on algebra or calculus (ie those that presuppose your level of knowlege).

The above function is an error. f(x)=(pi^3/8)(x-x^2) expanded in sine over0<x<1 gives a series that can be used sum=f(1/2)=pi^3/32 This function can be deduced by thinking about the sine expansion of 1 on the same interval and integrating twice with respect to x. Here it is with trilog(x) using a helpful identity from http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/17/02/03/01/ (trilog(1/z)-trilog(z))2=log^3(-z)/12+(pi^2)log(-z)/12 -i sum=(tilog(-i)-trilog(i))/2 -i sum=(tilog(1/i)-trilog(i))/2 -i sum=log^3(-i)/12+(pi^2)log(-i)/12 -i sum=(-pi i/2)^3/12+(pi^2)(-pi i/2)/12 -i sum=pi^3 i/96-pi^3 i /24 -i sum=-i pi^3/32 sum=pi^3/32

Easy way is subjective. I did find a function whose fourier series in cos that can be used to find the sum. f(x)=pi^3/32-(pi^3/16)x^2 0<x<.5 f(x)=pi^3/64-(pi^3/16)(x-.5)^2 .5<x<1 f(x)=(pi^3/16)(x-1)^2 1<x<1.5 f(x)=pi^3/64+(pi^3/16)(x-1.5)^2 1.5<x<2 find cos series the sum is f(0)=pi^3/32 A few easy things to try when finding sums 1. recognize it in terms of a known function my first thought when I saw it was sum=im(trilog(i)) but that was no easier than the sum, the integrals just gave back polylogs giving no progress. Next I thought of it as one of the functions related to zeta in fact Dirichlet Beta this gave the sum easily. The above function also gives the sum. 2. constuct a function have residues equal to terms of the sum so that the sum can be calculated by contour integration. That did not seem to good for this. 3. add a parameter then relate the sum to a known quantity and solve for it as in 1 I imbeded the sum in trilog(x) a 3rd order linear ode was apparent but knowing the solution was trilog(x) was not helpful. Any of those three approches make it clear why pi occurs in so many sums. I'm sure there are some good books on doing sums out there.

The sum is pi^3/32~.968946146259 It is the Dirichlet Beta Function evaluated at 3. http://mathworld.wolfram.com/DirichletBetaFunction.html

O.K. First realize that as niffty as this method is it will not work if without change if their are repeated of nonlinear factors. The idea is we know from a theorem in algebra that there exit numbers a,b,c so that (x-2)/[(x+1)(x+2)(x+4)]=a/(x+1)+b/(x+2)+c/(x+4) the cover up method works like this, take a number that makes the denominator zero such as -1 in this example. replace all the x's except the one in (x+1) with -1 (x'-2)/[(x+1)(x'+2)(x'+4)] x'->-1 (-1-2)/[(x+1)(-1+2)(-1+4)]=(-3)/[(x+1)(1)(3)]=-1/(x+1) do the same thing with -2 and -4 (-2-2)/[(-2+1)(x+2)(-2+4)]=(-4)/[(-1)(x+2)(2)]=2/(x+2) (-4-2)/[(-4+1)(-4+2)(x+4)]=(-6)/[(-3)(-2)(x+4)]=-1/(x+4) the sum of these three equals the original function (x-2)/[(x+1)(x+2)(x+4)]=-1/(x+1)+2/(x+2)-1/(x+4) This new form is much easier to integrate.

http://encyclopedia.laborlawtalk.com/Definitions_of_the_exponential_function A disapointing reference as it does not include my perfered definition exp(x) is the function (which exist and is unique) having the following 2 properties 1. exp(x+y)=exp(x)exp(y) for all x and y 2. lim [exp(x)-1]/x=1 x->0 (can be used for real or complex arguments)

But what I am trying to figure out, is where Euler got the idea to take the limit of (1+x/n)^n. That's such an odd thing to just do for no reason at all. Here is a good article on e=2.71828... 2.71828... I don't know where Euler got the idea, but there are good reason why one would think of it. Say you have a method for estimating exp(x) when x is small. For example exp(x)~1+x when x is small. x/n is small if n is large. exp(x)=(exp(x/n))^n~(1+x/n)^n if n is large.