Math - Page 6

In this section we’re continuing to discuss non-elementary functions called Euler integrals. These special functions are widely used in mathematics and physics. It was once discovered that they describe properties of elementary particles in the string theory. We won’t dig into this amazing topic today, though. For starters let’s talk about what is beta function and define its key properties for solving problems in math. Basically, later we’ll show you how these functions can simplify the evaluation process of certain pretty complicated integrals.

Here’s the Euler integral of the first kind, which is also known as beta function:

B(p,q)=\int_0^{1}{x^{p-1}(1-x)^{q-1}dx}

where p and q are parameters. Continue reading →

Today we’re going to discuss one of the non-elementary functions called gamma function and consider some of its properties. Gamma function is of great importance, it’s widely applied in math (in particular, when integrating certain types of expression gamma function helps greatly, we’ll see that later in examples), also gamma function is used in probability theory (possibly, you’ve heard about gamma distribution), etc. One of the most common representations of gamma function is the following:

\Gamma(p)=\int_0^{\infty}{e^{-x}x^{p-1}dx}, p>0

This is only true for positive values of parameter p. So to obtain value of gamma function in a certain point p we need to integrate over x the expression e^{-x} x^{(p-1)}, leaving p as a free parameter. Notice limits of integration – zero and infinity. This means that we’re dealing with improper integral. Continue reading →

Some time ago we’ve received a pretty interesting question from one of our visitors. It sounds as follows:

Let A be any set of 20 distinct integers chosen from the arithmetic progression 1, 4, 7,  …, 100.
Prove that there must be two distinct integers in A whose sum is 104. Continue reading →

In this section we’ll consider an example of how to deal with initial value problem (or Cauchy problem) for non-homogeneous second order differential equation with constant coefficients.

Initial value problem usually arises in the analysis of processes for which we know differential evolution law and the initial state. For example, consider the problem of counting human population. After years of observation and data processing  scientists came up with some differential equations with which we can describe number of human beings on Earth. This is called a population model, and actually there are several various approaches concerning this problem. Suppose we want to find out how many people will be there up to 2120 year. And here comes initial value problem. We know current population (our initial value) and have a differential equation, so to find future number of humans we’re to solve a Cauchy problem.

Ok, back to math. Continue reading →

In this section we offer one more example of how to solve system of linear algebraic equations using Gaussian elimination method. This example clearly shows that while doing Gaussian elimination you ought to notice when it’s convenient to swap rows in order to save time and reduce calculations. Continue reading →

In previous sections you can find theoretical background and examples including application of matrix representation and example of linear system having no solutions. As we already said, if you’re solving a system of linear equations there are three possible situations: there’s either unique solution, or infinitely many solutions, or no solution at all. Obviously, in your homework you potentially can obtain any of those when solving systems of linear equations, so that’s a nice idea to know how to deal with each of them. In this section we’re going to discuss the case when system of linear equations has infinitely many solutions. Continue reading →