APPLICATIONS OF ANALYTIC FUNCTIONS

AND COMPLEX INTEGRATION

Analytic function

• In mathematics , an  analyti c function  is a function  that is locally given by a  convergent power series . There exist both  real analytic functions and  complex analytic functions, categories that are similar in some ways, but different in others.

• Functions of each type are infinitely differentiable , but complex analytic functions exhibit properties that do not hold generally for real analytic functions

• Analytic functions are the last set of operations performed in a query except for the final ORDER BY clause. All joins and all WHERE, GROUP BY, and HAVING clauses are completed before the analytic functions are processed. Therefore, analytic functions can appear only in the select list or ORDER BY clause.

Applications of analytic functions

• Analytic functions are commonly used to compute cumulative, moving, centered, and reporting aggregates.

• To calculate employees under each manager

• The analytical function provides you the count(*) of each manager irrespective of the other column data selected by your query

• Adding a analytical function in your select clause is just a calculated value apart from whatever columns you select

• Functions of a complex variable provide us some powerful and widely useful tools in in theoretical physics.

• Some important physical quantities are complex variables

Complex integration

• Evaluating definite integrals.• Obtaining asymptotic solutions of differentials

equations. Integral transforms• Many Physical quantities that were originally

real become complex as simple theory is made more general. The energy

Applications

• These are the famous Cauchy-Riemann conditions. These Cauchy-Riemann conditions are necessary for the existence of a derivative, that is, if exists, the C-R conditions must hold.

• Conversely, if the C-R conditions are satisfied and the partial derivatives of u(x,y) and v(x,y) are continuous,exists.

Theorems used in complex integration

• Cauchy’s integral Theorem• Laurent Series We frequently encounter functions that

are analyticin annular region

• Taylor Expansion Suppose we are trying to expand f(z) about z=zand we have z=z 1 as the nearest point for

which f(z) is not analytic.

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