Complex analysis

Zeros and poles

In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point z0 is a pole of a function f if it is a zero of the function 1/f and 1/f is holomorphic in some neighbourhood of z0 (that is, complex differentiable in a neighbourhood of z0). 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 either f or 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 fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the multiplicities of its poles equals the sum of the multiplicities of its zeros. (Wikipedia).

Zeros and poles
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Poles and Zeros of z-Transforms

http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Definition of poles and zeros for z-transforms that are a ratio of polynomials in z. Examples.

From playlist The z-Transform

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What are zeros of a polynomial

👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number

From playlist Zeros and Multiplicity of Polynomials | Learn About

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Overview Zeros of a functions - Online Math Tutor - Free Math Videos

👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number

From playlist Zeros and Multiplicity of Polynomials | Learn About

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What do the zeros roots tell us of a polynomial

👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number

From playlist Zeros and Multiplicity of Polynomials | Learn About

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Overview of zeros of a polynomial - Online Tutor - Free Math Videos

👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number

From playlist Zeros and Multiplicity of Polynomials | Learn About

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Determine the Zeros of a Polynomial by Factoring

👉 Learn how to find all the zeros of a polynomial by grouping. A polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. The zeros of a polynomial are the values of x for which the value of the polynomial

From playlist Zeros of a Polynomial by Factoring

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Find the Zeros of a Polynomial by Factoring Substitution

👉 Learn how to find all the zeros of a polynomial. A polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. The zeros of a polynomial are the values of x for which the value of the polynomial is zero. T

From playlist Zeros of a Polynomial by Factoring

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Find the zeros factoring vs square root method

👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number

From playlist Zeros and Multiplicity of Polynomials | Learn About

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5. Root Locus

MIT Electronic Feedback Systems (1985) View the complete course: http://ocw.mit.edu/RES6-010S13 Instructor: James K. Roberge License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT Electronic Feedback Systems (1985)

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6. More Root Locus

MIT Electronic Feedback Systems (1985) View the complete course: http://ocw.mit.edu/RES6-010S13 Instructor: James K. Roberge License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT Electronic Feedback Systems (1985)

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Understanding and Sketching the Root Locus

In this video we discuss how to sketch the root locus for a system by developing a series of 5 core rules augmented by 5 supplemental rules (for a total of 10 rules). These rules will help us gain an understanding and intuition on how the root locus behaves as the parameter K increases fr

From playlist Control Theory

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Complex analysis: Classification of elliptic functions

This lecture is part of an online undergraduate course on complex analysis. We give 3 description of elliptic functions: as rational functions of P and its derivative, or in terms of their zeros and poles, or in terms of their singularities. We end by giving a brief description of the a

From playlist Complex analysis

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Lec 7 | MIT RES.6-008 Digital Signal Processing, 1975

Lecture 7: z-Transform properties Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES6-008S11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT RES.6-008 Digital Signal Processing, 1975

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EE102: Introduction to Signals & Systems, Lecture 17

These lectures are from the EE102, the Stanford course on signals and systems, taught by Stephen Boyd in the spring quarter of 1999. More information is available at https://web.stanford.edu/~boyd/ee102/

From playlist EE102: Introduction to Signals & Systems

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Introduction to Poles, Zeros, and the System Function

http://AllSignalProcessing.com for more great signal-processing content: ad-free videos, concept/screenshot files, quizzes, MATLAB and data files. Introduction to the relationship between the system function and and difference equation descriptions for linear time-invariant system. Defin

From playlist Introduction and Background

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What is multiplicity and what does it mean for the zeros of a graph

👉 Learn about zeros and multiplicity. The zeroes of a polynomial expression are the values of x for which the graph of the function crosses the x-axis. They are the values of the variable for which the polynomial equals 0. The multiplicity of a zero of a polynomial expression is the number

From playlist Zeros and Multiplicity of Polynomials | Learn About

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Complex analysis: Singularities

This lecture is part of an online undergraduate course on complex analysis. We discuss the different sorts of singularities of a holomorphic function (removable singularities, poles, essential singularities, branch-points, limits of singularities, natural boundaries) and give examples of

From playlist Complex analysis

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Frequency Response Magnitude and Poles and Zeros

http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Graphical interpretation of the magnitude response of a system described by a linear constant-coefficient difference equation in terms of the locati

From playlist The z-Transform

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Determine the Zeros for a Polynomial by Factoring

👉 Learn how to find all the zeros of a factored polynomial. A polynomial is an expression of the form ax^n + bx^(n-1) + . . . + k, where a, b, and k are constants and the exponents are positive integers. The zeros of a polynomial are the values of x for which the value of the polynomial is

From playlist Zeros of a Polynomial by Factoring

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Complex analysis: Elliptic functions

This lecture is part of an online undergraduate course on complex analysis. We start the study of elliptic (doubly periodic) functions by constructing some examples, and finding some conditions that their poles and zeros must satisfy. For the other lectures in the course see https://www

From playlist Complex analysis

Related pages

Hurwitz's theorem (complex analysis) | Rational function | Sendov's conjecture | Complex analysis | Zero of a function | Riemann hypothesis | Isolated point | Nyquist stability criterion | Differentiable function | Filter (signal processing) | Isomorphism | Residue (complex analysis) | Riemann sphere | Polynomial | Filter design | Point at infinity | Branch point | Meromorphic function | Neighbourhood (mathematics) | Complex plane | Laurent series | Pole–zero plot | Riemann surface | Gamma function | Multiplicity (mathematics) | Riemann–Roch theorem | Taylor series | Holomorphic function | Compact space | Analytic function | Gauss–Lucas theorem | Atlas (topology) | Rouché's theorem | Singularity (mathematics) | Riemann zeta function | Disk (mathematics) | Open set | Multiplicative inverse | Marden's theorem