A partial fractions decomposition can be useful if we want to compute an integral where the functions p(x) and q(x) are polynomials in x. This post explains, how such a decomposition works and how this method helps to simplify such integrals. At first we notice that the problem lies in the polynomial q(x) of theContinue reading “How to integrate by partial fractions”
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How to convert a complex number to polar form and back
There are three very common ways of writing any complex number z: Standard form: z = x + i y Polar form: z = |z| (cosφ + i sinφ) Exponential form: z = |z| eiφ In this post we explain how to get from any of these representations to the other ones. From standard formContinue reading “How to convert a complex number to polar form and back”
What is a cartesian product?
The cartesian product M1×M2 of two sets M1 and M2 is the set of all pairs (x1,x2) with x1∈M1 and x2∈M2. More generally, the cartesian product M1×M2×…×Mn of n sets M1,M2, …,Mn ist the set of all n-tuples (x1,x2,…,xn) with x1∈M1, x2∈M2, …, xn∈Mn. We may write Before we turn to some examples, let usContinue reading “What is a cartesian product?”
How to prove by induction
Mathematical induction is a method to systematically prove that a statement which depends on an integer is true for all values of this integer which are greater than some number. Let us be more precise and call S(n) a statement which involves an integer n (not to be confused with a function of n). AnContinue reading “How to prove by induction”
How to solve a system of linear equations
This post explains how to find the solutions of any system of linear equations with an arbitrary number of variables, if solutions exist. The method consists of the following three steps:
Using Gauss elimination we write the system in triangular form.
In this format, we see whether the system has either no solution, exactly one solution or infinitely many solutions.
If there are solutions, we determine them by simple insertions.
How to divide by a complex number
When we divide a complex number by another one, say we usually want to obtain the result in standard form. So how do we simplify a quotient such that the result takes the form ? The answer is: We multiply the numerator and the denominator of the quotient by the complex conjugate of theContinue reading “How to divide by a complex number”
math for physicists 