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The index of both curves around zero is therefore the same, so by the argument principle, f(z) and h(z) must have the same number of zeros inside C. One popular, informal way to summarize this argument is as follows: If a person were to walk a dog on a leash around and around a tree, such that the distance between the person and the tree is ...
A googol is the large number 10 100 or ten to the power of one hundred. In decimal notation, it is written as the digit 1 followed by one hundred zeros: ...
If the discriminant is equal to zero, there will be two coinciding real solutions at −b/2a. Finally, if the discriminant is greater than zero, there will be two real negative solutions, because b 2 − 4 a c < b {\displaystyle {\sqrt {b^{2}-4ac}}<b} for positive a , b and c .
Let f be a continuous function for which one knows an interval [a, b] such that f(a) and f(b) have opposite signs (a bracket). Let c = (a +b)/2 be the middle of the interval (the midpoint or the point that bisects the interval). Then either f(a) and f(c), or f(c) and f(b) have opposite
A polynomial f of degree n greater than one, which is irreducible over F q, defines a field extension of degree n which is isomorphic to the field with q n elements: the elements of this extension are the polynomials of degree lower than n; addition, subtraction and multiplication by an element of F q are those of the polynomials; the product ...
For example, given a = f(x) = a 0 x 0 + a 1 x 1 + ··· and b = g(x) = b 0 x 0 + b 1 x 1 + ···, the product ab is a specific value of W(x) = f(x)g(x). One may easily find points along W(x) at small values of x, and interpolation based on those points will yield the terms of W(x) and the specific product ab. As fomulated in Karatsuba ...
A strictly proper transfer function is a transfer function where the degree of the numerator is less than the degree of the denominator. The difference between the degree of the denominator (number of poles) and degree of the numerator (number of zeros) is the relative degree of the transfer function.
A function f is meromorphic in an open set U if for every point z of U there is a neighborhood of z in which at least one of f and 1/f is holomorphic. If f is meromorphic in U, then a zero of f is a pole of 1/f, and a pole of f is a zero of 1/f. This induces a duality between zeros and poles, that is