Complex number roots

If the roots are, however, complex numbers, with one or two positive real parts, the system response will diverge with time in an oscillatory manner, since the analytical solution is then one involving sine and cosine terms. If both roots, however, have negative real parts, the sine and cosine terms still cause an oscillatory response, but the oscillation will decay with time, back to the original steady-state value, which, therefore remains a stable steady state. 

A complex number consists of two parts a real and a so-called imaginary part, c = a + ib. The imaginary part always contains the quantity i, which represents the square root of -1, i = /—1- The real and imaginary parts of c are often denoted by a = R(c) and b = 1(c). All the common rules of ordinary arithmetic apply to complex numbers, which in addition allow extraction of the square root of any negative number. If... 

Most of the graphs we deal with are trees- that is, they do not have any self tracing loops. Choose a vertex which will be the root . In the example we consider here we shall take vertex 5 as the root (see figure l.(c)). A wave which propagates from the root along a certain branch is reflected, and this reflection can be expressed by a reflection from the vertex which is next to the root. Once we know the reflection coefficient, which because of unitarity is a complex number with unit modulus, we can construct the SB(k) matrix. In the present cases, when the valency of the root is 3, we get... 

The roots are now complex numbers with real and imaginary parts. 

If the roots and S2 are both real numbers, Bq. (6.116) shows that Uq and a I are certainly both real. If the roots and. S are complex, the coefficients and di must still be real and must also satisfy Eq. (6.116). Complex conjugates are the only complex numbers that give real numbers when they are multiplied together and when added together. To illustrate this, let z be a complex number z = X + iy. Let z be the complex conjugate of z z = Jt — iy. Now... 

Rules I to 4 are fairly self-evident. Rule 5 comes from the fact that at a point on the root locus plot the complex number s must satisfy the equation /... 

When a system has poles that are widely different in value, it is difficult to plot them all on a root locus plot using conventional rectangular coordinates in the s plane. U is sometimes more convenient to make the root locus plots in the log s plane. Instead of using the conventional axis Re s and Im s, an ordinate of the arg s and an abscissa of the log s arc used, since the natural logarithm of a complex number is defined ... 

These steady states are within the physically possible range of T 0 < T < oo) and X(0 < X < 1). This is in contrast to many situations in the physical sciences where equations have multiple roots but only one root is physically acceptable because the other solutions are either outside the bounds of parameters (such as negative concentrations or temperatures) or occur as imaginary or complex numbers and can therefore be ignored. 

We denote the set of complex numbers by C. Readers should be familiar with complex numbers and how to add and multiply them, as described in many standard calculus texts. We use i to denote the square root of —1 and an asterisk to denote complex conjugation if x and y are real numbers, then (x + iy) = X — iy. Later in the text, we will use the asterisk to denote the conjugate transpose of a matrix with complex entries. This is perfectly consistent if one thinks of a complex number x + iy as a one-by-one complex matrix ( x + iy ). See also Exercise 1.6. The absolute value of a complex number, also known as the modulus, is denoted... 

Neither of equations (iii) or (iv) are solutions to equation (7.46). However, if n was such that n2 was negative, then both functions would be solutions to the equation. This would require us to define the square root of a negative number, which is at odds with our understanding of what constitutes a real number. In Chapter 2, Volume 2, we extend the concept of the number to include so-called imaginary and complex numbers, which embrace the idea that the square root of a negative number can be defined. 

Here the root finder can be any one of the algorithms from the vast literature. Assuming that the chosen algorithm works properly, it takes the n real or complex coefficients a for i = 0 to i = n — 1 of p as input and produces n real or complex numbers Xj, the roots of p, as output. To do so, we have tacitly assumed that the leading coefficient an of p is normalized to equal 1 since dividing p of degree n by a constant (an 0) does not alter its roots. 

Learn to plot complex numbers on the complex plane by calling x = roots([1 -18 144 -672 2016 -4032 5376 -4608 2304 -512]),for example, first and then plotting the real and imaginary parts of the output vector x, as well as x = 2. Use the symbol to plot the entries of x and the + symbol for x =2. Where do the computed roots he in the complex plane How can you describe their location Also evaluate abs(x-2) what do these numbers mean and signify ... 

The PFs gi gy] are a set ofg2 complex numbers, which by convention are all chosen to be square roots of unity. (For vector representations the PFs are all unity.) PFs have the following properties (Altmann (1977)) ... 

Proof, (i) Since C is assumed to be the field of complex numbers, C is algebraically closed. Thus, x(as) is the sum of x(cri) characteristic roots of [Pg.196]

The complex number z is a primitive nth root of unity if zn=l but zk is not equal to 1 for any positive integer k less than n. probability... 

Polynomials A polynomial in z, a z" 4- a iz" 4— 4- flo. where n is a positive integer, is simply a sum of complex numbers times integral powers of z which have already been defined. Every polynomial of degree n has precisely n complex roots provided each multiple root of multiplicity m. is counted m. times. 

The set of roots of a polynomial with real coefficients can include conjugate pairs of complex numbers. Thus eigenvalues can be complex, appearing in conjugate pairs. When this happens the corresponding column eigenvectors also form a conjugate pair, as do the rows. 

It is possible to use the idea of evaluating the polynomial at a real or complex value as an aid to proving that one polynomial is a factor of another, by showing that all the roots of the first are also roots of the second. In fact in the Laplace transform the domain is definitely that of complex numbers, and [CDM91] uses this interpretation very fluently and to good effect. 

Use the Euler formula and the De Moivre theorem to evaluate powers of complex numbers, to determine / th roots of a complex number, and to identify real and imaginary parts of functions of a complex variable... 

In the answer to Worked Problem 2.1, we obtained the required roots of the quadratic equation in the form of a sum of a real number (-1) and an imaginary number (2i or -2i). Such numbers are termed complex numbers, and have the general form ... 

It follows that one square root of a complex number (where = ) is given by ... 

Finding the roots of positive, negative and complex numbers. 

Operations on complex numbers all start with IM, and use text strings to squeeze the two components of a complex number into one cell. In order to use the results of complex number operations, you must therefore first extract its real and imaginary components, using IMREALO and IMAGINARY () Instead of i you can use j to denote the square root of minus one (which you must then specify as such), but you cannot use the corresponding capitals, I or /. 

The fundamental theorem of algebra states that a polynomial of degree n has n number of roots, although some of the roots may be complex numbers. If a root of a polynomial of degree n is known to be, say r, then a polynomial of degree n - 1 is given by... 

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