৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]

Non-constant analytic functions cannot have a cluster of zeros accumulating inside the domain (Identity Theorem). Hence zeros are isolated.

By the Isolated Zeros Theorem, zeros of a non-constant analytic function are isolated points unless the function is identically zero.

A contour integral is of the form ∫_Cf(z) dz, where C is a curve in the complex plane. Analytic function theory allows evaluating such integrals using theorems like Cauchy’s theorem and Residue theorem.

For large ∣z∣, the leading term zⁿ dominates. By Rouche’s theorem, P(z)) has as many zeros as zⁿ, i.e., n zeros inside the circle of radius R.

Factor: P(z)=(z+1)(z+2). Roots: −1,−2. Both lie outside ∣z∣<1. So 0 zeros inside.

Rouche’s theorem compares two analytic functions f(z) and g(z) on a contour and guarantees they have the same number of zeros inside if ∣g(z)∣<∣f(z)∣ on the contour.

P(z)=(z−2)³ has the single root z=2, but with multiplicity 3. Still satisfies Fundamental Theorem of Algebra, as total number of roots = degree = 3.

Direct result of Cauchy’s Theorem.

Taylor series: only non-negative powers, valid inside a disk where f is analytic.
Laurent series: allows negative powers in an annulus around isolated singularities.