১.
- কSimple zero
- খZero of order 1
- গZero of order 2
- ঘEssential singularity
৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]
মোট প্রশ্ন৩৫
সিলেবাস
Exam - 10
Topics:
Complex Analysis
(a) Complex Integration: Zeros of analytic functions, Cauchy’s theorem, Morera’s
theorem, Cauchy’s integral formula, Singularities, Classification of singularities.
(b) Complex integration: The open mapping theorem, Taylor’s and Laurent series.
Fundamental theorem of algebra, Rouches theorem, The residue theorem, Contour
integration.
[Source: Class - 07 and Relevant Books]
৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১]
৪৯তম বিসিএস ⎯ ফলিত গণিত [৫৬১] · তারিখ অনির্ধারিত · ৩৫ প্রশ্ন
১.
ব্যাখ্যা
২.
The point z = 0 is a removable singularity of which function?
ব্যাখ্যা
৩.
Which of the following has a pole?
ব্যাখ্যা
৪.
Evaluate
ব্যাখ্যা
৫.
Zeros of a non-constant analytic function are:
ব্যাখ্যা
Non-constant analytic functions cannot have a cluster of zeros accumulating inside the domain (Identity Theorem). Hence zeros are isolated.
৬.
ব্যাখ্যা
৭.
Identify the type of singularity of
ব্যাখ্যা
৮.
If f(z) = z2(z − 1), the number of zeros inside a contour enclosing z = 0 and z = 1 is:
৯.
Which singularity does
have at z=0?
have at z=0?
ব্যাখ্যা
১০.
What type of singularity doeshave at z=0?
ব্যাখ্যা
১১.
For
the singularity at z = 2 is:
the singularity at z = 2 is:
ব্যাখ্যা
১২.
If f(z) is non-constant and analytic in a domain D, which of the following is true?
ব্যাখ্যা
১৩.
The residue theorem is used to evaluate:
ব্যাখ্যা
১৪.
Which of the following statements is true for zeros of analytic functions?
ব্যাখ্যা
By the Isolated Zeros Theorem, zeros of a non-constant analytic function are isolated points unless the function is identically zero.
১৫.
If f(z) has a simple pole at z=a, then Res[f(z),a]=
ব্যাখ্যা
১৬.
Morera’s theorem is essentially the converse of which theorem?
ব্যাখ্যা
১৭.
Evaluate
ব্যাখ্যা
১৮.
If f(z) = z4 + 4, then the number of zeros and their order is:
ব্যাখ্যা
১৯.
Which of the following integrals equals zero?
ব্যাখ্যা
২০.
Contour integration evaluates:
ব্যাখ্যা
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.
২১.
Evaluate
ব্যাখ্যা
২২.
ব্যাখ্যা
২৩.
Evaluate
ব্যাখ্যা
২৪.
If P(z) = zn + (lower degree terms), then for sufficiently large ∣z∣ = R, the number of zeros inside the circle is:
ব্যাখ্যা
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.
২৫.
How many zeros of P(z) = z4 + 5z + 2 lie inside ∣z∣ = 1?
ব্যাখ্যা
২৬.
Use Rouche’s theorem to find the number of zeros of P(z) = z2 + 3z + 2 inside ∣z∣ = 1.
ব্যাখ্যা
Factor: P(z)=(z+1)(z+2). Roots: −1,−2. Both lie outside ∣z∣<1. So 0 zeros inside.
২৭.
Evaluate
ব্যাখ্যা
২৮.
Fundamental Theorem of Algebra
ব্যাখ্যা
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.
২৯.
The polynomial P(z) = (z − 2)3 has:
ব্যাখ্যা
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.
৩০.
A cubic polynomial P(z) = z3 - 1 has how many roots in C?
ব্যাখ্যা
৩১.
If f(z) is analytic in a simply connected domain D, then for any closed contour C in D
ব্যাখ্যা
Direct result of Cauchy’s Theorem.
৩২.
Which statement is correct?
ব্যাখ্যা
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.
৩৩.
If f(z)=ez and is expanded about z0=0, which is the correct Taylor series?
ব্যাখ্যা
৩৪.
Evaluate
ব্যাখ্যা
৩৫.