Q3
(a) Locate the poles and their order for the function f(z) = 1/[z(sin πz)(z + 1/2)]. Also, find the residue of f(z) at these poles. (15 marks) (b) Consider the series Σ(n=1 to ∞) U_n(x), 0 ≤ x ≤ 1, the sum of whose first n terms is given by S_n(x) = (1/2n²)log(1 + n⁴x²), x ∈ [0,1]. Show that the given series can be differentiated term-by-term, though Σ(n=1 to ∞) U'_n(x), does not converge uniformly on [0,1]. (20 marks) (c) Using duality principle, solve the following linear programming problem: Minimize z = 4x₁ + 3x₂ + x₃ subject to x₁ + 2x₂ + 4x₃ ≥ 12, 3x₁ + 2x₂ + x₃ ≥ 8, x₁, x₂, x₃ ≥ 0. (15 marks)
हिंदी में प्रश्न पढ़ें
(a) फलन f(z) = 1/[z(sin πz)(z + 1/2)] के अनंतक तथा उनकी घात (ऑर्डर) का पता लगाइए। इन अनंतकों पर f(z) के अवशेष भी ज्ञात कीजिए। (15 अंक) (b) श्रेणी Σ(n=1 to ∞) U_n(x), 0 ≤ x ≤ 1 का विचार कीजिए, जिसके पहले n पदों का योगफल S_n(x) = (1/2n²)log(1 + n⁴x²), x ∈ [0,1] के द्वारा दिया गया है। दर्शाइए कि दी गई श्रेणी को पद-दर-पद अवकलित किया जा सकता है, यद्यपि Σ(n=1 to ∞) U'_n(x), [0,1] पर एकसमान अभिसरित नहीं होती है। (20 अंक) (c) द्वैत सिद्धांत का उपयोग करते हुए, निम्नलिखित रैखिक प्रोग्रामन समस्या को हल कीजिए: न्यूनतमीकरण कीजिए z = 4x₁ + 3x₂ + x₃ बशर्ते कि x₁ + 2x₂ + 4x₃ ≥ 12, 3x₁ + 2x₂ + x₃ ≥ 8, x₁, x₂, x₃ ≥ 0। (15 अंक)
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How this answer will be evaluated
Approach
Solve this multi-part numerical problem by allocating approximately 30% time to part (a) on complex analysis, 40% to part (b) on series differentiation, and 30% to part (c) on linear programming via duality. Begin each part with clear identification of the mathematical technique required, proceed through systematic computation with explicit formula citations, and conclude with verified numerical answers for residues, convergence justification, and optimal primal/dual values.
Key points expected
- For (a): Identify all poles at z = 0, z = -1/2, and z = n (n ∈ ℤ, n ≠ 0) with correct orders; calculate residues using Laurent series or limit formulas, especially handling the simple pole at z = -1/2 and double pole at z = 0
- For (a): Correctly classify z = 0 as a double pole (order 2) due to the combined effect of z and sin(πz), and simple poles at z = n for non-zero integers
- For (b): Derive U_n(x) = S_n(x) - S_{n-1}(x), show term-by-term differentiation is valid by verifying uniform convergence of ΣU_n(x) and conditions for differentiation under the series
- For (b): Demonstrate that ΣU'_n(x) does not converge uniformly on [0,1] by analyzing the behavior at x = 0 versus x > 0, particularly showing sup-norm of partial sums does not tend to zero
- For (c): Formulate the dual LP correctly (maximize w = 12y₁ + 8y₂ subject to y₁ + 3y₂ ≤ 4, 2y₁ + 2y₂ ≤ 3, 4y₁ + y₂ ≤ 1, y₁, y₂ ≥ 0), solve using graphical or simplex method, and apply complementary slackness to recover primal optimal solution
- For (c): Verify strong duality by confirming equal optimal values and interpret shadow prices for the primal constraints
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | Correctly identifies all singularities in (a) with proper order classification; properly sets up U_n(x) = S_n - S_{n-1} in (b); accurately formulates dual LP with correct inequality directions and objective in (c) | Identifies most poles but misclassifies order at z=0; sets up U_n with minor errors; dual formulation has sign or constraint errors that are partially correctable | Misses essential singularities, confuses pole orders, or fails to set up dual; fundamental setup errors prevent meaningful progress |
| Method choice | 20% | 10 | Selects appropriate residue formulas (simple pole formula, Laurent expansion for double pole); chooses correct convergence theorems for term-by-term differentiation; applies duality principle with complementary slackness efficiently | Uses standard formulas but with suboptimal choices (e.g., unnecessary Laurent expansion for simple poles); attempts correct theorems but with gaps in application; solves dual but with inefficient method | Uses incorrect residue formulas, confuses uniform vs pointwise convergence criteria, or attempts primal solution instead of duality approach |
| Computation accuracy | 20% | 10 | Computes all residues correctly including Res(f,0) = 2/π, Res(f,-1/2) = -2/π; accurately derives U_n(x) and U'_n(x); obtains correct dual optimal y₁=0, y₂=0.5 and primal x₁=0, x₂=2.5, x₃=1.75 with z=9.25 | Correct residue at simple poles but error in double pole calculation; correct form of U_n but algebraic slips in limit evaluation; correct dual solution but arithmetic errors in back-substitution | Major computational errors in residues, incorrect derivative calculations, or fundamentally wrong optimal values due to calculation mistakes |
| Step justification | 20% | 10 | Explicitly justifies why z=0 is order 2 using sin(πz) ~ πz; proves non-uniform convergence of ΣU'_n by showing sup|Σ_{k=n}^{2n} U'_k(x)| ≥ c > 0; verifies complementary slackness conditions rigorously | States reasons for pole orders without full justification; asserts non-uniform convergence without complete ε-N argument; applies complementary slackness with minor logical gaps | Lacks justification for key steps, asserts convergence properties without proof, or applies duality mechanically without verification |
| Final answer & units | 20% | 10 | Presents complete residue table with values at all poles; clearly states term-by-term differentiation is valid while ΣU'_n fails uniform convergence; gives exact optimal solution with minimum z = 37/4 or 9.25 and all variable values | Most answers present but incomplete (missing some residues or unclear on convergence distinction); correct optimal value but missing some variable values | Missing critical final answers, incorrect boxed results, or failure to distinguish between primal and dual optimal values |
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