What is regular point in complex analysis?

What is regular point in complex analysis?

a mathematical term used in different senses. A regular point of a function f(z) of the complex variable z is a point z0 = x0 + iy0 such that in some neighborhood ǀz – z0ǀ < ρ of it the function is single-valued and can be represented in the form of the series.

What are regular and irregular points?

Regular singular points are well-behaved and defined in terms of the ratio Q(x)/P(x) and R(x)/P(x), where P(x), Q(x), and R(x) are the polynomial coefficients in the differential equation you’re trying to solve. Irregular singular points are a totally different ball game — and one that I don’t get into in this chapter.

What is regular singular point with example?

Example: x2y + 2(ex − 1)y + e−x cos xy = 0, P = x2, Q = 2(ex − 1), R = e−x cos x. x = 0 is a singular point. Since the quotient functions p = xQ/P and q = x2R/P have Taylor Expansions about x = 0, x = 0 is a regular singular point.

What is a regular point of a function?

In other words, a regular point for a function Rn → R is a point at which at least one of the partials is not zero. By adding the condition that we are at a regular point, we can obtain a partial converse to the Implicit Function Theorem in §§8.1 and 8.2. on any neighborhood of P0.

What is ordinary point?

Definition. A point x0 is an ordinary point if both P(x) and Q(x) are analytic. at x0. If a point in not ordinary it is a singular point.

What is regular point in optimization?

Regular point of a set of constraints: A feasible vector x for which the constraint gradients {∇h1(x), ··· , ∇hm(x)} are linearly independent. For a local minimum that is not regular, there may not exist Lagrange multipliers. minimize f(x) = x1 + x2, s.t.

What is analytic point?

Definition: A function f is called analytic at a point z0 ∈ C if there exist r > 0 such that f is differentiable at every point z ∈ B(z0, r). A function is called analytic in an open set U ⊆ C if it is analytic at each point U.

What is ordinary point in series solution?

A point x0 is said to be an ordinary point of the differential equation y + P(x)y + Q(x)y = 0 if both P(x) and Q(x) are analytic at x0. A point is singular if it is not ordinary.

How do you know which singular point is regular?

Suppose that and are polynomials with no common factors. If after reducing and to lowest terms, the highest power of in the denominator of is 1 and the highest power of in the denominator of is 2, then is a regular singular point of the equation.

Which feasible points are regular?

A point x* satisfying the constraints h(x*) = 0 is said to be a regular point of the feasible set if f(x*) is differentiable and gradient vectors of all constraints at the point x* are linearly independent.

What is non negativity constraint?

Non-negative constraints: Each decision variable in any Linear Programming model must be positive irrespective of whether the objective function is to maximize or minimize the net present value of an activity.

What is meant by singular point?

singular point in American English noun. Math. a point at which a given function of a complex variable has no derivative but of which every neighborhood contains points at which the function has derivatives. Also called: singularity.

What is meant by regular singular points?

Definition 4.13 Singular points of equations with polynomial coefficients. Suppose that and are polynomials with no common factors. If after reducing and to lowest terms, the highest power of in the denominator of is 1 and the highest power of in the denominator of is 2, then is a regular singular point of the equation …

What is meant by feasible region?

A feasible region is an area defined by a set of coordinates that satisfy a system of inequalities. The region satisfies all restrictions imposed by a linear programming scenario. The concept is an optimization technique.

What is a corner point?

The corner points (or extreme points) of a feasible region are the points of intersection between two (or more) constraints. A feasible region may be bounded or unbounded but shall have at least one corner point.

What is called feasible solution?

A solution (set of values for the decision variables) for which all of the constraints in the Solver model are satisfied is called a feasible solution. In some problems, a feasible solution is already known; in others, finding a feasible solution may be the hardest part of the problem.