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Browse College Analysis

Stars indicate particularly interesting answers or good places to begin browsing.



The Impossibility of Integrating x^x [08/07/1998]
How would you test to see if the integral of x^x can be expressed in a finite number of elementary functions?

Indefinite Integral [6/1/1996]
I want to find ( | e^(x*x) dx "e to the x squared").

Infinite Series Involving Arithmetic and Geometric Means [11/27/2003]
Start with two numbers, find their arithmetic and geometric means, then find the means of the two results and continue this process indefinitely. Does the series converge?

Intermediate Value Theorem [09/19/2002]
How can we prove by the intermediate value theorem that there is a point on the path that a hiker will cross at exactly the same time of the day hiking up and returning?

Is the Number Line Both Continuous and Porous? [05/13/2005]
The number line is said to be densely populated because between any two numbers are infinite other numbers. But aren't there also an infinite number of gaps in the number line, with a tiny gap on either side of each of those discrete infinite numbers?

Leftmost Digits of 2^n [11/01/2002]
Prove that there is a power of 2 whose decimal representation starts with the digits 1999.

Limit of x sin(1/x) [04/23/2002]
I assumed from the graph that the function had a limit at x=0 of 0, but since it involves sin(1/0) I can not prove this using the basic trigonometric limits (sin x/x and (1-cos x)/x), L'Hopital's rule, or by rearranging the equation. Can you help?

Limit Superior and Limit Inferior [03/27/2005]
I'm reading a real analysis text and I'm having trouble understanding completely what lim sup and lim inf are with regard to the notation and the concept in general. Can you explain them?

Linear Interpolation Methods to Approximate Roots [12/25/2000]
What is the best criterion to use for ending the iteration process of linear interpolation methods for approximating non-linear equation roots? Also, which is the better explanation of the Newton-Raphson method; the geometric method, or using Taylor's expansion?

The Logarithm of a Complex Number [07/11/2007]
I'm wondering how to calculate the log of a complex number. More specifically, I need to calculate arccosh(x), where 'x' can be any negative number.

Mathematics and Philosophy [05/15/1999]
Is 2 + 3 really equal to 5?

Meaning of R++ [9/12/1996]
What do the symbols R_{+} and R_{++} mean?

Measure Theory and Sigma Algebras [03/24/2003]
I'm trying to understand what a 'measure' is.

Metric Spaces [12/11/2002]
Let (X,d) be a metric space with the discrete metric. Prove that no subset of X has a limit point.

Minimizing the Length of a Crease [11/11/1999]
How can I minimize the the length (L) of the crease if one of the corners of an 8.5" x 11" sheet of paper is lifted up and placed on the opposite longer edge, and then the paper is smoothed flat?

Overview of Riemann's Zeta Function and Prime Numbers [04/11/2006]
Can you please give an overview of the importance of the Zeta function and finding prime numbers? Why is the Zeta function such a hot topic in the field of looking for prime numbers?

Polarized Tensors [04/30/2003]
What is a polarized tensor, and how can we prove that it has rank two? With a list of relevant books on the subject of mathematical tensors.

Problem from Real Analysis [10/05/2002]
Let X = A U B where A and B are subspaces of X. Let f:X->Y. Suppose that the restricted functions f|A:A->Y and f|B:B->Y are continuous. Show that if A and B are closed in X, then f is continuous.

Product of Negative Numbers [02/04/1997]
Can you give a proof of why the product of two negative numbers is positive?

Proof of Roots of Odd Degree Polynomial [05/10/2000]
How can I prove that any real polynomial mapping of odd degree has a root?

Proofs of e [03/21/2002]
I found out that the definition of e is: e = lim, as n approaches infinity (1+1/n)^n. Is there a proof for this or for 1+2(e-1/n)?

Prove... pi [7/5/1996]
Prove that the sum of 1/n^2 with n-> {1 to infinity} is <=> to pi^2/6.

Prove S' Closed in R^n [10/07/2002]
Prove that S' (the derived set or set of all accumulation points of S) is closed in R^n.

Prove That a Set Is Uncountably Infinite [10/31/1997]
Prove that the set [0,1]x[0,1] is uncountably infinite.

Proving a Set is Closed [04/15/1998]
Let f be a continuous real valued function on Rn, and let S = {x in Rn : f(x) = 1}. Prove that S is a closed subset of Rn.

Proving Limits at Infinity Using the Formal Definition [04/11/2004]
We are required to find a limit (informally), then prove that our answer is correct by using the formal definition of limits. I'm not sure I'm on the right track.

Proving the Properties of Natural Numbers [03/08/2000]
How can you prove or derive the commutative, associative, and distributive properties of numbers?

Rational/Irrational Numbers [03/12/1999]
Prove that between any two numbers there are a rational and an irrational number and, therefore, that there are infinite numbers of each.

Real Analysis vs. Complex Analysis [11/15/1997]
My students want to know why one has to study real analysis at all if one can study complex anlaysis, since they feel one could just study any question in real analysis with the tools of complex analysis by assuming the imaginary part 0...

Real Plane, Complex Plane [09/16/2002]
When does Az + Bz + c = 0 become a straight line?

Riemann Integrals, etc. [4/24/1995]
Let A be a subset of the Reals, prove that: i) If A is finite, then the outer measure is zero, and ii) If A = [a,b], then the outer measure = (b- a)...

The Riemann Zeta Function: Extended Confusion about an Analytic Continuation [11/11/2011]
A non-native English speaker plugs negative values into the Riemann Zeta Function, and struggles to reconcile the results. Doctor Vogler clarifies a common misconception, which comes from the function's oft misunderstood definition.

Riemann Zeta Hypothesis [09/28/1997]
I have been puzzled by the zeta function and its relation to the prime number theory...

A Sigma Algebra that Contains Intervals (a,infinity) Contains Every Interval [09/21/2010]
A student wants to prove that a sigma algebra containing all intervals of the form (a,infinity) in fact contains all intervals. Doctor Jordan turns to the complement of a known interval to get both himself and the student unstuck.

Size of Infinite Sets [11/20/1996]
Which is larger: the number of parabolas, or the number of hyperbolas?

Solving the Equation x^y = y^x [12/09/2004]
Solve x^y = y^x for x in terms of y only. Also, how do I find all possible solutions beyond the obvious ones of x = y, (2,4), and (4,2)?

Transcendental Equation x = a^x [03/17/2004]
Given y1(x) = x and y2(x) = A^x, where A is a constant real number between 0 and 1, what is the analytic expression for the intersection of y1(x) and y2(x) in terms of A?

Two Definitions of Limits, with Examples [05/11/1998]
Epsilon-delta definitions of the limit of a function and the limit of a sequence.

An Uncountable Set [09/28/1998]
How do you prove, using diagonalization, that the set of functions from N to N is uncountable?

Uniform Convergence of a Sequence [8/16/1995]
I assert that uniform convergence of a sequence of functions on a given interval to a function, f, is a "stronger" statement than pointwise convergence of the sequence. Is my thinking correct?

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