Prove or disprove b The integral on the unit circle of a fun

Prove or disprove

b. The integral on the unit circle of a function that is not analytic at 0 cannot equal 0.

Solution

The absolute value function when defined on the set of real numbers or complex numbers is not everywhere analytic because it is not differentiable at 0. Piecewise defined functions (functions given by different formulas in different regions) are typically not analytic where the pieces meet.

a) the disc would not be analytic at z=i , so it might not be able to attain the minimum absolute value at z = i .

the disk is not analytic al the point z=i . We\'ll have to disapprovr with this statement.

(b)

an analytic 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. A function is analytic if and only if its Taylor series about x₀ converges to the function in some neighborhood for every x₀ in its domain.

The integral on the unit circle of a function will not be analytic at 0 . This statement is true.

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