No matter how much we may try to ignore it, human communication always takes place in a context, through a medium, and among individuals and groups who are situated historically, politically, economically, and socially. This state of affairs is neither bad nor good. Bias is a small word that identifies the collective influences of the entire context of a message. Politicians are certainly biased and overtly so.
As a result, the value of a holomorphic function over an arbitrarily small region in fact determines the value of the function everywhere to which it can be extended as a holomorphic function.
Major results[ edit ] One of the central tools in complex analysis is the line integral. The line integral around a closed path of a function that is holomorphic everywhere inside the area bounded by the closed path is always zero, which is what the Cauchy integral theorem states.
The values of such a holomorphic function inside a disk can be computed by a path integral on the disk's boundary as shown in Cauchy's integral formula.
Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory of residues among others is applicable see methods of contour integration.
A "pole" or isolated singularity of a function is a point where the function's value becomes unbounded, or "blows up". If a function has such a pole, then one can compute the function's residue there, which can be used to compute path integrals involving the function; this is the content of the powerful residue theorem.
The remarkable behavior of holomorphic functions near essential singularities is described by Picard's Theorem. Functions that have only poles but no essential singularities are called meromorphic.
Laurent series are the complex-valued equivalent to Taylor seriesbut can be used to study the behavior of functions near singularities through infinite sums of more well understood functions, such as polynomials. A bounded function that is holomorphic in the entire complex plane must be constant; this is Liouville's theorem.
It can be used to provide a natural and short proof for the fundamental theorem of algebra which states that the field of complex numbers is algebraically closed. If a function is holomorphic throughout a connected domain then its values are fully determined by its values on any smaller subdomain.
The function on the larger domain is said to be analytically continued from its values on the smaller domain. This allows the extension of the definition of functions, such as the Riemann zeta functionwhich are initially defined in terms of infinite sums that converge only on limited domains to almost the entire complex plane.
Sometimes, as in the case of the natural logarithmit is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface.
All this refers to complex analysis in one variable. There is also a very rich theory of complex analysis in more than one complex dimension in which the analytic properties such as power series expansion carry over whereas most of the geometric properties of holomorphic functions in one complex dimension such as conformality do not carry over.
The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane, which may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions.This is an extremely literate and somewhat scholarly look at the past, present, and future of the emerging art form of interactive narrative, where storytelling, visual imagery, and reader interaction meet.
Students are asked to write literary analysis essays because this type of assignment encourages you to think about how and why a poem, short story, novel, or play was written. To successfully analyze literature, you’ll need to remember that authors make specific choices for particular reasons.
Analysis: Narrative Point of View SUMMARIZE THE STORY IN A FEW SENTENCES.
1 Narrative Analysis CATHERINE KOHLER RIESSMAN narrative form, but analysis of formal properties is not attempted. Carole Cain goes a bit further () in her study of identity acquisition abstract (summary and/or point of . Log in to view your courses, explore tools and features, and customize your eLearning experience.
First Time Here? Run a system check on your computer to make sure. Media / Political Bias. There is no such thing as an objective point of view. No matter how much we may try to ignore it, human communication always takes place in a context, through a medium, and among individuals and groups who are situated historically, politically, economically, and socially.