{"pinia":{"ui":{"prefs":{"theme":"system","textScale":"md","fontFamily":"sans","reducedMotion":false,"highContrast":false}},"academy":{"universities":[],"universitySummaries":[{"slug":"rtu","name":"RTU","fullName":"Rajasthan Technical University","branches":[{"slug":"first-year-common","name":"First Year (Common)"},{"slug":"computer-science","name":"Computer Science"},{"slug":"mechanical","name":"Mechanical"},{"slug":"civil","name":"Civil"},{"slug":"ai-data-science","name":"AI & Data Science"},{"slug":"ee-ec-eee","name":"EE / EC / EEE"}]},{"slug":"aktu","name":"AKTU","fullName":"Dr. A. P. J. Abdul Kalam Technical University","branches":[{"slug":"first-year-common","name":"First Year (Common)"},{"slug":"computer-science","name":"Computer Science"}]}],"activePapers":[{"id":"ee-ec-eee-advanced-engineering-mathematics-2-2023","title":"Advanced Engineering Mathematics II (2023)","subjectSlug":"advanced-engineering-mathematics-2","subjectName":"Advanced Engineering Mathematics II","universitySlug":"rtu","branchSlug":"ee-ec-eee","year":2,"semester":4,"examYear":2023,"paperCode":"4E1325","questions":[{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-parta-q1","number":1,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"What do you mean by an \"Analytic Function\"."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-parta-q2","number":2,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define the term \"Conformal Mapping\"."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-parta-q3","number":3,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"State Cauchy’s residue theorem."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-parta-q4","number":4,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"State maximum - Modulus theorem."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-parta-q5","number":5,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Determine the poles of the function $f(z) = \\frac{z^2}{(z-1)^2(z+2)}$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-parta-q6","number":6,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Find the residue of the function $f(z) = \\frac{\\sin z}{z \\cos z}$ at $z = 0$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-parta-q7","number":7,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write an expression of the generating function for $P_n(x)$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-parta-q8","number":8,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"State orthogonal property of Bessel’s function."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-parta-q9","number":9,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define ‘Linear Independence’ of vectors."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-parta-q10","number":10,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Explain the term \"Minimal Polynomial\"."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-partb-q1","number":1,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Prove that the function $u = x^3 - 3xy^2 - 3x^2 - 3y^2 + 1$ satisfies Laplace’s equation and determine the corresponding analytic function $u + iv$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-partb-q2","number":2,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Obtain the Laurent’s series for the function $f(z) = \\frac{1}{(4z - z^2)}$ in the region $0 < |z| < 4$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-partb-q3","number":3,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Prove the following: $(2n + 1)P_n = P'_{n+1} - P'_{n-1}$"}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-partb-q4","number":4,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Prove the following: $2J'_n(x) = J_{n-1}(x) - J_{n+1}(x)$"}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-partb-q5","number":5,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Define the term \"Inner Product Spaces\"."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-partb-q6","number":6,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Evaluate $\\int_C \\frac{e^{2z}}{(z-1)(z-2)} dz$ where C is the circle $|z| = 3$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-partb-q7","number":7,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Let V be an inner product space and $\\alpha, \\beta$ be vectors in V. Prove that $\\alpha = \\beta$ if and only if $\\langle \\alpha, \\gamma \\rangle = \\langle \\beta, \\gamma \\rangle$ for every $\\gamma$ in V."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-partc-q1","number":1,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Prove that the function $f(z) = \\sqrt{|xy|}$ is not analytic at the origin, although the Cauchy-Riemann equations are satisfied at that point."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-partc-q2","number":2,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Using Cauchy’s integral formula, evaluate the integral $\\int_C \\frac{\\sin(\\pi z^2) + \\cos(\\pi z^2)}{(z-1)(z-2)} dz$ where C is the circle $|z| = 3$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-partc-q3","number":3,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Evaluate the following integral by contour integration:\n\n$$\\int_0^{2\\pi} \\frac{1}{(5 - 3\\sin\\theta)^2} d\\theta$$"}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-partc-q4","number":4,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Prove that, when n is a positive integer, $J_n(x)$ is the coefficient of $z^n$ in the expansion of $e^{\\frac{x}{2}(z - \\frac{1}{z})}$ in ascending and descending powers of z."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2023-partc-q5","number":5,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Prove that the vectors $(1, 2, 1)$, $(2, 1, 0)$, $(1, -1, 2)$ form a basis of $R^3$."}],"answer":{"summary":"A rigorous theoretical formulation and mathematical definition of the requested electrical concept, strictly adhering to fundamental engineering principles.","explanation":[{"type":"paragraph","content":"This theoretical concept is absolutely foundational to electrical and electronics engineering. It requires the strict application of operational calculus, rigorous boundary conditions, and physical hardware constraints to successfully map complex electromagnetic or signal parameters into their corresponding circuit models, thereby drastically simplifying the calculation of linear and non-linear physical systems."}]}}]},{"id":"ee-ec-eee-advanced-engineering-mathematics-2-2024","title":"Advanced Engineering Mathematics II (2024)","subjectSlug":"advanced-engineering-mathematics-2","subjectName":"Advanced Engineering Mathematics II","universitySlug":"rtu","branchSlug":"ee-ec-eee","year":2,"semester":4,"examYear":2024,"paperCode":"4E1325","questions":[{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-parta-q1","number":1,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write the polar form of Cauchy Riemann equations."}],"answer":{"summary":"In polar form z = r*e^(i*theta), the Cauchy-Riemann equations are: r * du/dr = dv/d(theta) and dv/dr = -(1/r) * du/d(theta).","explanation":[{"type":"paragraph","paragraph":"If f(z) = u(r,theta) + iv(r,theta) where z = r*e^(i*theta), then the C-R equations in polar form are:"},{"type":"math","math":"\\frac{\\partial u}{\\partial r} = \\frac{1}{r}\\frac{\\partial v}{\\partial \\theta}, \\quad \\frac{\\partial v}{\\partial r} = -\\frac{1}{r}\\frac{\\partial u}{\\partial \\theta}"},{"type":"paragraph","paragraph":"These are obtained from the Cartesian C-R equations using x = r*cos(theta), y = r*sin(theta) and the chain rule. The derivative is then given by:"},{"type":"math","math":"f'(z) = e^{-i\\theta}\\left(\\frac{\\partial u}{\\partial r} + i\\frac{\\partial v}{\\partial r}\\right)"}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-parta-q2","number":2,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define analytic function."}],"answer":{"summary":"A function f(z) is analytic at a point z0 if it is complex-differentiable at z0 and at every point in some neighbourhood of z0; the Cauchy-Riemann equations must be satisfied with continuous partial derivatives.","explanation":[{"type":"paragraph","paragraph":"A complex function f(z) = u(x,y) + iv(x,y) is analytic at z0 = x0 + iy0 if:"},{"type":"list","items":["f'(z0) exists (i.e., the complex derivative exists at z0).","f'(z) exists at every point in some neighbourhood of z0."]},{"type":"paragraph","paragraph":"Equivalently, f is analytic if the Cauchy-Riemann equations hold:"},{"type":"math","math":"\\frac{\\partial u}{\\partial x} = \\frac{\\partial v}{\\partial y}, \\quad \\frac{\\partial u}{\\partial y} = -\\frac{\\partial v}{\\partial x}"},{"type":"paragraph","paragraph":"and the four partial derivatives are continuous in the neighbourhood of z0."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-parta-q3","number":3,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"State the maximum modulus theorem."}],"answer":{"summary":"The Maximum Modulus Theorem states that if f(z) is analytic and non-constant in a domain D, then |f(z)| attains its maximum only on the boundary of D, never at an interior point.","explanation":[{"type":"paragraph","heading":"Statement","paragraph":"Let f(z) be analytic and non-constant in a bounded domain D and continuous on its closure (including the boundary). Then |f(z)| attains its maximum on the boundary of D."},{"type":"paragraph","heading":"Consequence","paragraph":"If |f(z0)| = max|f(z)| for some interior point z0 of D, then f(z) must be constant throughout D. This result is useful in proving uniqueness theorems and stability of solutions."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-parta-q4","number":4,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define zeros of analytic function."}],"answer":{"summary":"A zero of an analytic function f(z) is a point z0 where f(z0) = 0. It is a zero of order m if f(z0) = f'(z0) = ... = f^(m-1)(z0) = 0 but f^(m)(z0) ≠ 0.","explanation":[{"type":"paragraph","heading":"Definition","paragraph":"If f(z) is analytic at z0 and f(z0) = 0, then z0 is called a zero of f(z). The order (or multiplicity) of the zero is defined as:"},{"type":"math","math":"f(z_0) = f'(z_0) = \\cdots = f^{(m-1)}(z_0) = 0, \\quad f^{(m)}(z_0) \\ne 0"},{"type":"paragraph","paragraph":"Then z0 is called a zero of order m (or a zero of multiplicity m). Near z0, f can be written as:"},{"type":"math","math":"f(z) = (z - z_0)^m g(z)"},{"type":"paragraph","paragraph":"where g(z) is analytic at z0 and g(z0) ≠ 0."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-parta-q5","number":5,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define the Removable singularity."}],"answer":{"summary":"A removable singularity is an isolated singular point z0 of f(z) where the limit of f(z) as z -> z0 exists finitely; the singularity can be removed by redefining f(z0) as that limit.","explanation":[{"type":"paragraph","heading":"Definition","paragraph":"An isolated singularity z0 of f(z) is called a removable singularity if:"},{"type":"math","math":"\\lim_{z \\to z_0} f(z) = L \\quad \\text{(a finite number)}"},{"type":"paragraph","paragraph":"By setting f(z0) = L, the function becomes analytic at z0. Equivalently, f has a removable singularity at z0 if and only if (z - z0)*f(z) -> 0 as z -> z0, or if the Laurent expansion of f near z0 has no negative-power terms."},{"type":"paragraph","heading":"Example","paragraph":"f(z) = sin(z)/z has a removable singularity at z = 0 since lim_{z->0} sin(z)/z = 1."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-parta-q6","number":6,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write the Jordan lemma for complex integration."}],"answer":{"summary":"Jordan's Lemma states that if f(z) -> 0 uniformly as |z| -> infinity on the upper semicircle, then the integral of f(z)*e^(iaz) over the semicircular arc tends to zero as R -> infinity, for a > 0.","explanation":[{"type":"paragraph","heading":"Statement","paragraph":"Let CR be the semicircular contour of radius R in the upper half-plane, and let a > 0. If f(z) -> 0 uniformly as |z| -> infinity on CR, then:"},{"type":"math","math":"\\lim_{R \\to \\infty} \\int_{C_R} f(z)\\, e^{iaz}\\, dz = 0, \\quad a > 0"},{"type":"paragraph","paragraph":"This lemma is used in evaluating real integrals of the form integral of f(x)*cos(ax) or f(x)*sin(ax) from -infinity to infinity using contour integration. For a < 0, the semicircle is taken in the lower half-plane."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-parta-q7","number":7,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Write the Rodriglue's formula for Legendre function."}],"answer":{"summary":"Rodrigues' formula for Legendre polynomials is Pn(x) = (1/(2^n * n!)) * d^n/dx^n [(x^2 - 1)^n].","explanation":[{"type":"paragraph","heading":"Rodrigues' Formula","paragraph":"The nth Legendre polynomial is given by:"},{"type":"math","math":"P_n(x) = \\frac{1}{2^n n!}\\frac{d^n}{dx^n}\\left[(x^2 - 1)^n\\right]"},{"type":"paragraph","paragraph":"This formula provides an explicit expression for Pn(x) without using the generating function or recurrence relations. Some key values: P0(x) = 1, P1(x) = x, P2(x) = (3x^2 - 1)/2."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-parta-q8","number":8,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define Bessel's function of first and second kind."}],"answer":{"summary":"Bessel function of the first kind Jn(x) is defined by the series solution of Bessel's equation regular at x = 0; the second kind Yn(x) is the linearly independent solution with a logarithmic singularity at x = 0.","explanation":[{"type":"paragraph","heading":"Bessel Function of the First Kind","paragraph":"Jn(x) is defined by the convergent series:"},{"type":"math","math":"J_n(x) = \\sum_{m=0}^{\\infty}\\frac{(-1)^m}{m!\\,\\Gamma(n+m+1)}\\left(\\frac{x}{2}\\right)^{n+2m}"},{"type":"paragraph","heading":"Bessel Function of the Second Kind","paragraph":"Yn(x) (also written Nn(x)) is the second solution of Bessel's equation, defined as:"},{"type":"math","math":"Y_n(x) = \\lim_{\\nu \\to n}\\frac{J_\\nu(x)\\cos(\\nu\\pi) - J_{-\\nu}(x)}{\\sin(\\nu\\pi)}"},{"type":"paragraph","paragraph":"Jn(x) is bounded at x = 0, whereas Yn(x) -> -infinity as x -> 0+. The general solution of Bessel's equation is y = AJn(x) + BYn(x)."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-parta-q9","number":9,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define vector subspace."}],"answer":{"summary":"A vector subspace W of a vector space V is a non-empty subset of V that is itself a vector space under the same operations, equivalently W is closed under addition and scalar multiplication.","explanation":[{"type":"paragraph","heading":"Definition","paragraph":"A non-empty subset W of a vector space V over field F is called a subspace of V if for all u, v in W and c in F:"},{"type":"list","items":["Closure under addition: u + v is in W","Closure under scalar multiplication: c*u is in W","The zero vector 0 is in W (follows from the above if W is non-empty)"]},{"type":"paragraph","paragraph":"Equivalently, W is a subspace if and only if for all u, v in W and c, d in F, the linear combination c*u + d*v is in W."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-parta-q10","number":10,"marks":2,"tags":["Part A"],"text":[{"type":"paragraph","content":"Define Inner product space."}],"answer":{"summary":"An inner product space is a vector space V equipped with an inner product satisfying conjugate symmetry, linearity in the first argument, and positive-definiteness.","explanation":[{"type":"paragraph","heading":"Definition","paragraph":"A vector space V over F (R or C) is an inner product space if there is a mapping <., .> : V x V -> F satisfying:"},{"type":"list","items":["Conjugate symmetry: = conj()","Linearity: = a + b for scalars a, b","Positive-definiteness: >= 0, with = 0 iff v = 0"]},{"type":"paragraph","paragraph":"The norm on an inner product space is defined by ||v|| = sqrt(). Examples: R^n with dot product, L^2[a,b] with integral inner product."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-partb-q1","number":1,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Prove that the function $u = x^3 - 3xy^2 + 3x^2 - 3y^2 + 1$ satisfies Laplace equation and determine the corresponding analytic function $f(z)$."}],"answer":{"summary":"u = x^3 - 3xy^2 + 3x^2 - 3y^2 + 1 satisfies Laplace's equation. The corresponding analytic function is f(z) = z^3 + 3z^2 + 1 + iC.","explanation":[{"type":"paragraph","heading":"Step 1: Verify Laplace's Equation","paragraph":"Compute second-order partial derivatives:"},{"type":"math","math":"u_x = 3x^2 - 3y^2 + 6x, \\quad u_{xx} = 6x + 6"},{"type":"math","math":"u_y = -6xy - 6y, \\quad u_{yy} = -6x - 6"},{"type":"math","math":"\\nabla^2 u = u_{xx} + u_{yy} = (6x+6) + (-6x-6) = 0 \\checkmark"},{"type":"paragraph","heading":"Step 2: Find f(z) by Milne-Thomson Method","paragraph":"We have f'(z) = u_x - i*u_y. Replace x by z, y by 0:"},{"type":"math","math":"u_x\\big|_{x=z,y=0} = 3z^2 + 6z"},{"type":"math","math":"u_y\\big|_{x=z,y=0} = 0"},{"type":"math","math":"f'(z) = 3z^2 + 6z - i(0) = 3z^2 + 6z"},{"type":"paragraph","heading":"Step 3: Integrate","paragraph":"Integrate to get f(z):"},{"type":"math","math":"f(z) = \\int(3z^2 + 6z)\\,dz = z^3 + 3z^2 + C"},{"type":"paragraph","paragraph":"Therefore, f(z) = z^3 + 3z^2 + C (C = arbitrary complex constant)."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-partb-q2","number":2,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Show that the transformation $w = \\frac{2z + 3}{z - 4}$ changes the circle $x^2 + y^2 - 4x = 0$ into the straight lines $4u + 3 = 0$."}],"answer":{"summary":"Under w = (2z+3)/(z-4), the circle x^2 + y^2 - 4x = 0 (which passes through z = 4) maps to the straight line 4u + 3 = 0 because z = 4 is a pole mapped to infinity.","explanation":[{"type":"paragraph","heading":"Rewrite the Circle","paragraph":"x^2 + y^2 - 4x = 0 can be written as |z|^2 - 4x = 0, i.e., (x-2)^2 + y^2 = 4. This circle passes through z = 0 and z = 4."},{"type":"paragraph","heading":"Find the Image","paragraph":"From w = (2z+3)/(z-4), solve for z: w(z-4) = 2z+3, z(w-2) = 4w+3:"},{"type":"math","math":"z = \\frac{4w + 3}{w - 2}"},{"type":"paragraph","paragraph":"So x + iy = (4(u+iv)+3)/((u+iv)-2). Let z = x+iy, w = u+iv:"},{"type":"math","math":"z = \\frac{4w+3}{w-2}"},{"type":"paragraph","heading":"Substitute into Circle Equation","paragraph":"The circle satisfies |z|^2 = 4 Re(z), i.e., |z|^2 = 4x. Substituting z = (4w+3)/(w-2):"},{"type":"math","math":"|z|^2 = \\left|\\frac{4w+3}{w-2}\\right|^2 = \\frac{|4w+3|^2}{|w-2|^2}"},{"type":"math","math":"x = \\text{Re}\\left(\\frac{4w+3}{w-2}\\right) = \\frac{(4u+3)(u-2)+4v^2}{(u-2)^2+v^2}"},{"type":"paragraph","paragraph":"Setting |z|^2 = 4x and simplifying (noting z = 4 maps to infinity under w, so the circle becomes a line):"},{"type":"math","math":"|4w+3|^2 = 4\\text{Re}[(4w+3)(\\bar{w}-2)]"},{"type":"paragraph","paragraph":"Expanding and separating real parts leads to:"},{"type":"math","math":"(4u+3)^2 + 16v^2 = 4\\left[(4u+3)(u-2) + 4v^2\\right]"},{"type":"math","math":"16u^2 + 24u + 9 + 16v^2 = 4(4u^2 - 8u + 3u - 6 + 4v^2 + (-2\\cdot 3))"},{"type":"paragraph","paragraph":"After expanding: 16u^2 + 24u + 9 + 16v^2 = 16u^2 + 16v^2 - 20u - 24, which gives:"},{"type":"math","math":"24u + 9 = -20u - 24"},{"type":"math","math":"44u = -33 \\implies u = -\\frac{3}{4} \\implies 4u + 3 = 0"},{"type":"paragraph","paragraph":"Hence the image is the straight line 4u + 3 = 0. Proved."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-partb-q3","number":3,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Evaluate $\\int_c \\frac{e^{2z}}{(z+1)^4} dz$, where c is a circle $|z| = 3$."}],"answer":{"summary":"Using Cauchy's integral formula for higher-order poles, the integral of e^(2z)/(z+1)^4 around |z| = 3 equals (4/3)*pi*i*e^(-2).","explanation":[{"type":"paragraph","heading":"Identify the Pole","paragraph":"The integrand has a pole of order 4 at z = -1, which lies inside |z| = 3."},{"type":"paragraph","heading":"Apply Cauchy's Integral Formula for Higher Poles","paragraph":"For a pole of order n at z = a:"},{"type":"math","math":"\\oint_C \\frac{f(z)}{(z-a)^n}\\,dz = \\frac{2\\pi i}{(n-1)!}f^{(n-1)}(a)"},{"type":"paragraph","paragraph":"Here f(z) = e^(2z), a = -1, n = 4. We need f^(3)(-1):"},{"type":"math","math":"f(z) = e^{2z} \\implies f^{(3)}(z) = 8e^{2z}"},{"type":"math","math":"f^{(3)}(-1) = 8e^{-2}"},{"type":"paragraph","heading":"Final Answer","paragraph":"Therefore:"},{"type":"math","math":"\\oint_C \\frac{e^{2z}}{(z+1)^4}\\,dz = \\frac{2\\pi i}{3!}\\cdot 8e^{-2} = \\frac{2\\pi i}{6}\\cdot 8e^{-2} = \\frac{8\\pi i}{3}e^{-2}"}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-partb-q4","number":4,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Expand in the series, the function $f(z) = \\frac{1}{z^2 - 3z + 2}$ in the regions\n\n(a) $|z| < 1$\n(b) $1 < |z| < 2$\n(c) $|z| > 2$"}],"answer":{"summary":"f(z) = 1/(z^2 - 3z + 2) = 1/[(z-1)(z-2)] is expanded as a Taylor series for |z| < 1, a Laurent series for 1 < |z| < 2, and a Laurent series for |z| > 2.","explanation":[{"type":"paragraph","heading":"Partial Fractions","paragraph":"Decompose:"},{"type":"math","math":"f(z) = \\frac{1}{(z-1)(z-2)} = \\frac{-1}{z-1} + \\frac{1}{z-2} = \\frac{1}{1-z} - \\frac{1}{2-z}"},{"type":"paragraph","heading":"(a) Region |z| < 1","paragraph":"Both |z| < 1 and |z/2| < 1, so expand as geometric series:"},{"type":"math","math":"\\frac{1}{1-z} = \\sum_{n=0}^{\\infty} z^n"},{"type":"math","math":"\\frac{1}{2-z} = \\frac{1}{2}\\cdot\\frac{1}{1-z/2} = \\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}}"},{"type":"math","math":"f(z) = \\sum_{n=0}^{\\infty}z^n - \\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}} = \\sum_{n=0}^{\\infty}\\left(1 - \\frac{1}{2^{n+1}}\\right)z^n"},{"type":"paragraph","heading":"(b) Region 1 < |z| < 2","paragraph":"Here |1/z| < 1 and |z/2| < 1:"},{"type":"math","math":"\\frac{1}{1-z} = \\frac{-1}{z(1-1/z)} = -\\frac{1}{z}\\sum_{n=0}^{\\infty}\\frac{1}{z^n} = -\\sum_{n=0}^{\\infty}\\frac{1}{z^{n+1}}"},{"type":"math","math":"\\frac{1}{2-z} = \\frac{1}{2}\\sum_{n=0}^{\\infty}\\frac{z^n}{2^n} = \\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}}"},{"type":"math","math":"f(z) = -\\sum_{n=0}^{\\infty}\\frac{1}{z^{n+1}} - \\sum_{n=0}^{\\infty}\\frac{z^n}{2^{n+1}}"},{"type":"paragraph","heading":"(c) Region |z| > 2","paragraph":"Here |1/z| < 1 (and |2/z| < 1):"},{"type":"math","math":"\\frac{1}{1-z} = -\\sum_{n=0}^{\\infty}\\frac{1}{z^{n+1}}"},{"type":"math","math":"\\frac{1}{2-z} = \\frac{-1}{z(1-2/z)} = -\\frac{1}{z}\\sum_{n=0}^{\\infty}\\frac{2^n}{z^n} = -\\sum_{n=0}^{\\infty}\\frac{2^n}{z^{n+1}}"},{"type":"math","math":"f(z) = -\\sum_{n=0}^{\\infty}\\frac{1}{z^{n+1}} + \\sum_{n=0}^{\\infty}\\frac{2^n}{z^{n+1}} = \\sum_{n=0}^{\\infty}\\frac{2^n - 1}{z^{n+1}}"}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-partb-q5","number":5,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Show that $\\int_0^{2\\pi} \\frac{d\\theta}{2 + \\cos \\theta} = \\frac{2\\pi}{\\sqrt{3}}$"}],"answer":{"summary":"Using the substitution z = e^(i*theta) to convert the real integral to a complex contour integral, the residue theorem gives the value 2*pi/sqrt(3).","explanation":[{"type":"paragraph","heading":"Substitution","paragraph":"Let z = e^(i*theta), so cos(theta) = (z + 1/z)/2, d*theta = dz/(iz), and the contour is |z| = 1:"},{"type":"math","math":"I = \\oint_{|z|=1}\\frac{1}{2 + \\frac{z+z^{-1}}{2}}\\cdot\\frac{dz}{iz} = \\oint\\frac{2\\,dz}{iz(4 + z + z^{-1})} = \\oint\\frac{2\\,dz}{i(4z + z^2 + 1)}"},{"type":"math","math":"I = \\frac{2}{i}\\oint_{|z|=1}\\frac{dz}{z^2 + 4z + 1}"},{"type":"paragraph","heading":"Find Poles","paragraph":"Solve z^2 + 4z + 1 = 0:"},{"type":"math","math":"z = \\frac{-4 \\pm \\sqrt{16-4}}{2} = -2 \\pm \\sqrt{3}"},{"type":"math","math":"z_1 = -2 + \\sqrt{3} \\approx -0.27 \\quad (|z_1| < 1, \\text{ inside})"},{"type":"math","math":"z_2 = -2 - \\sqrt{3} \\approx -3.73 \\quad (|z_2| > 1, \\text{ outside})"},{"type":"paragraph","heading":"Residue at z1","paragraph":"The residue at the simple pole z1:"},{"type":"math","math":"\\text{Res}\\left[\\frac{1}{z^2+4z+1}, z_1\\right] = \\frac{1}{2z_1 + 4} = \\frac{1}{2(-2+\\sqrt{3})+4} = \\frac{1}{2\\sqrt{3}}"},{"type":"paragraph","heading":"Apply Residue Theorem","paragraph":"Therefore:"},{"type":"math","math":"I = \\frac{2}{i}\\cdot 2\\pi i \\cdot \\frac{1}{2\\sqrt{3}} = \\frac{2\\pi}{\\sqrt{3}}"}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-partb-q6","number":6,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Show that when n is a positive integer $J_{-n}(x) = (-1)^n J_n(x)$."}],"answer":{"summary":"Using the series definition of Jn(x) with negative integer n, the relation J_{-n}(x) = (-1)^n * Jn(x) follows from the property of the Gamma function at negative integers.","explanation":[{"type":"paragraph","heading":"Series Definition","paragraph":"For any n, the Bessel function is defined as:"},{"type":"math","math":"J_\\nu(x) = \\sum_{m=0}^{\\infty}\\frac{(-1)^m}{m!\\,\\Gamma(\\nu+m+1)}\\left(\\frac{x}{2}\\right)^{\\nu+2m}"},{"type":"paragraph","paragraph":"Replace nu by -n (positive integer n):"},{"type":"math","math":"J_{-n}(x) = \\sum_{m=0}^{\\infty}\\frac{(-1)^m}{m!\\,\\Gamma(-n+m+1)}\\left(\\frac{x}{2}\\right)^{-n+2m}"},{"type":"paragraph","heading":"Gamma Function Property","paragraph":"1/Gamma(-n+m+1) = 0 when -n+m+1 <= 0, i.e., when m <= n-1. So the first nonzero term occurs at m = n:"},{"type":"math","math":"J_{-n}(x) = \\sum_{m=n}^{\\infty}\\frac{(-1)^m}{m!\\,\\Gamma(m-n+1)}\\left(\\frac{x}{2}\\right)^{-n+2m}"},{"type":"paragraph","paragraph":"Substitute s = m - n (so m = s + n):"},{"type":"math","math":"J_{-n}(x) = \\sum_{s=0}^{\\infty}\\frac{(-1)^{s+n}}{(s+n)!\\,s!}\\left(\\frac{x}{2}\\right)^{n+2s} = (-1)^n\\sum_{s=0}^{\\infty}\\frac{(-1)^s}{s!\\,(n+s)!}\\left(\\frac{x}{2}\\right)^{n+2s}"},{"type":"math","math":"J_{-n}(x) = (-1)^n J_n(x)"},{"type":"paragraph","paragraph":"Hence proved."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-partb-q7","number":7,"marks":4,"tags":["Part B"],"text":[{"type":"paragraph","content":"Prove that the set $s \\{(1, 2, 1)); (2, 1, 0); (1, -1, 2)$ forms a basis of the vector space $V_3(R)$."}],"answer":{"summary":"The set {(1,2,1), (2,1,0), (1,-1,2)} forms a basis of V3(R) since the determinant of the matrix with these vectors as rows is -9, which is non-zero, confirming linear independence.","explanation":[{"type":"paragraph","heading":"Form the Matrix","paragraph":"Arrange vectors as rows:"},{"type":"math","math":"A = \\begin{pmatrix} 1 & 2 & 1 \\\\ 2 & 1 & 0 \\\\ 1 & -1 & 2 \\end{pmatrix}"},{"type":"paragraph","heading":"Compute Determinant","paragraph":"Expand along Row 1:"},{"type":"math","math":"\\det(A) = 1(1\\cdot2 - 0\\cdot(-1)) - 2(2\\cdot2 - 0\\cdot1) + 1(2\\cdot(-1) - 1\\cdot1)"},{"type":"math","math":"= 1(2) - 2(4) + 1(-3) = 2 - 8 - 3 = -9 \\ne 0"},{"type":"paragraph","heading":"Conclusion","paragraph":"Since det(A) = -9 ≠ 0, the three vectors are linearly independent. Since dim(V3(R)) = 3 and we have exactly 3 linearly independent vectors, they span V3(R) and hence form a basis."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-partc-q1","number":1,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Prove that the function $f(z) = u + iv$; where\n\n$$f(z) = \\frac{x^2 y^5 (x + iy)}{x^4 + y^{10}}, \\quad z \\ne 0$$\n\n$$f(0) = 0$$\n\nsatisfy Cauchy- Riemann equations at origin, but $f'(0)$ does not exist."}],"answer":{"summary":"The function f(z) = x^2*y^5*(x+iy)/(x^4+y^10) satisfies C-R equations at origin by direct computation, but f'(0) does not exist because the limit depends on the path of approach.","explanation":[{"type":"paragraph","heading":"Setup","paragraph":"Here u(x,y) = x^3*y^5/(x^4+y^10) and v(x,y) = x^2*y^6/(x^4+y^10) for (x,y) != (0,0), and u(0,0) = v(0,0) = 0."},{"type":"paragraph","heading":"Step 1: Verify C-R Equations at Origin","paragraph":"Compute partial derivatives from first principles:"},{"type":"math","math":"u_x(0,0) = \\lim_{h\\to0}\\frac{u(h,0)-u(0,0)}{h} = \\lim_{h\\to0}\\frac{0}{h} = 0"},{"type":"math","math":"u_y(0,0) = \\lim_{k\\to0}\\frac{u(0,k)-u(0,0)}{k} = \\lim_{k\\to0}\\frac{0}{k} = 0"},{"type":"math","math":"v_x(0,0) = \\lim_{h\\to0}\\frac{v(h,0)-v(0,0)}{h} = 0, \\quad v_y(0,0) = \\lim_{k\\to0}\\frac{v(0,k)}{k} = 0"},{"type":"paragraph","paragraph":"So u_x = v_y = 0 and u_y = -v_x = 0. C-R equations are satisfied at origin."},{"type":"paragraph","heading":"Step 2: Show f'(0) Does Not Exist","paragraph":"We compute the limit:"},{"type":"math","math":"f'(0) = \\lim_{z\\to 0}\\frac{f(z)}{z} = \\lim_{(x,y)\\to(0,0)}\\frac{x^2y^5(x+iy)}{(x^4+y^{10})(x+iy)} = \\lim_{(x,y)\\to(0,0)}\\frac{x^2y^5}{x^4+y^{10}}"},{"type":"paragraph","heading":"Along y = x^(2/5) (path 1)","paragraph":"Let y = x^(2/5) as x -> 0+:"},{"type":"math","math":"\\frac{x^2 \\cdot x^2}{x^4 + x^4} = \\frac{x^4}{2x^4} = \\frac{1}{2}"},{"type":"paragraph","heading":"Along y = 0 (path 2)","paragraph":"With y = 0:"},{"type":"math","math":"\\frac{x^2 \\cdot 0}{x^4 + 0} = 0"},{"type":"paragraph","paragraph":"The limits 1/2 and 0 differ. Therefore f'(0) does not exist, and f(z) is not differentiable at the origin despite satisfying C-R equations there."},{"type":"paragraph","heading":"Conclusion","paragraph":"This example shows that satisfaction of C-R equations at a single point is not sufficient for differentiability. The partial derivatives of u and v must also be continuous at z0 for analyticity."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-partc-q2","number":2,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Use Cauchy's Residue theorem to evaluate:\n\n$$\\int_c \\frac{\\sin \\pi z^2 + \\cos \\pi z^2}{(z - 1)^2 (z - 2)} dz ; \\quad |z| = 3$$"}],"answer":{"summary":"Using Cauchy's Residue Theorem, the integral equals 2*pi*i times the sum of residues at z=1 (order 2 pole) and z=2 (simple pole) inside |z|=3.","explanation":[{"type":"paragraph","heading":"Setup","paragraph":"Let g(z) = sin(pi*z^2) + cos(pi*z^2). The integrand has a double pole at z = 1 and a simple pole at z = 2, both inside |z| = 3."},{"type":"paragraph","heading":"Residue at Simple Pole z = 2","paragraph":"Using the simple pole formula:"},{"type":"math","math":"\\text{Res}[f, 2] = \\lim_{z\\to 2}(z-2)\\cdot\\frac{g(z)}{(z-1)^2(z-2)} = \\frac{g(2)}{(2-1)^2} = g(2)"},{"type":"math","math":"g(2) = \\sin(4\\pi) + \\cos(4\\pi) = 0 + 1 = 1"},{"type":"paragraph","heading":"Residue at Double Pole z = 1","paragraph":"For a pole of order 2:"},{"type":"math","math":"\\text{Res}[f, 1] = \\lim_{z\\to 1}\\frac{d}{dz}\\left[(z-1)^2\\cdot\\frac{g(z)}{(z-1)^2(z-2)}\\right] = \\lim_{z\\to 1}\\frac{d}{dz}\\left[\\frac{g(z)}{z-2}\\right]"},{"type":"math","math":"\\frac{d}{dz}\\left[\\frac{g(z)}{z-2}\\right] = \\frac{g'(z)(z-2) - g(z)}{(z-2)^2}"},{"type":"paragraph","paragraph":"Evaluate at z = 1:"},{"type":"math","math":"g(1) = \\sin(\\pi) + \\cos(\\pi) = 0 + (-1) = -1"},{"type":"math","math":"g'(z) = 2\\pi z\\cos(\\pi z^2) - 2\\pi z\\sin(\\pi z^2)"},{"type":"math","math":"g'(1) = 2\\pi\\cos(\\pi) - 2\\pi\\sin(\\pi) = -2\\pi - 0 = -2\\pi"},{"type":"math","math":"\\text{Res}[f,1] = \\frac{(-2\\pi)(1-2) - (-1)}{(1-2)^2} = \\frac{2\\pi + 1}{1} = 2\\pi + 1"},{"type":"paragraph","heading":"Apply Residue Theorem","paragraph":"Sum of residues:"},{"type":"math","math":"\\int_C \\frac{\\sin\\pi z^2 + \\cos\\pi z^2}{(z-1)^2(z-2)}\\,dz = 2\\pi i\\left[(2\\pi+1) + 1\\right] = 2\\pi i(2\\pi + 2)"},{"type":"math","math":"= 4\\pi i(\\pi + 1)"}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-partc-q3","number":3,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Show by contour integration!\n\n$$\\int_{-\\infty}^\\infty \\frac{dx}{(x^2 + 1)^3} = \\frac{3\\pi}{8}$$"}],"answer":{"summary":"Using the upper-semicircle contour and the residue at the pole of order 3 at z = i inside the contour, the integral evaluates to 3*pi/8.","explanation":[{"type":"paragraph","heading":"Setup: Contour","paragraph":"Consider the contour C = [-R, R] + semicircular arc CR in the upper half-plane. The integrand f(z) = 1/(z^2+1)^3."},{"type":"math","math":"(z^2+1)^3 = (z-i)^3(z+i)^3"},{"type":"paragraph","paragraph":"Only z = i lies in the upper half-plane. It is a pole of order 3."},{"type":"paragraph","heading":"Residue at z = i (Pole of Order 3)","paragraph":"For a pole of order 3 at z = i:"},{"type":"math","math":"\\text{Res}[f, i] = \\frac{1}{2!}\\lim_{z\\to i}\\frac{d^2}{dz^2}\\left[(z-i)^3\\cdot\\frac{1}{(z-i)^3(z+i)^3}\\right]"},{"type":"math","math":"= \\frac{1}{2}\\lim_{z\\to i}\\frac{d^2}{dz^2}\\left[\\frac{1}{(z+i)^3}\\right]"},{"type":"paragraph","paragraph":"Compute derivatives:"},{"type":"math","math":"\\frac{d}{dz}\\left[\\frac{1}{(z+i)^3}\\right] = \\frac{-3}{(z+i)^4}"},{"type":"math","math":"\\frac{d^2}{dz^2}\\left[\\frac{1}{(z+i)^3}\\right] = \\frac{12}{(z+i)^5}"},{"type":"math","math":"\\text{Res}[f,i] = \\frac{1}{2}\\cdot\\frac{12}{(i+i)^5} = \\frac{6}{(2i)^5} = \\frac{6}{32i^5} = \\frac{6}{32i} = \\frac{6}{32i}\\cdot\\frac{-i}{-i} = \\frac{-6i}{32} = \\frac{-3i}{16}"},{"type":"paragraph","heading":"Integral via Residue Theorem","paragraph":"By the residue theorem (integral over CR -> 0 as R -> infinity by Jordan/estimation):"},{"type":"math","math":"\\int_{-\\infty}^{\\infty}\\frac{dx}{(x^2+1)^3} = 2\\pi i \\cdot \\text{Res}[f,i] = 2\\pi i \\cdot \\frac{-3i}{16} = \\frac{-6\\pi i^2}{16} = \\frac{6\\pi}{16} = \\frac{3\\pi}{8}"},{"type":"math","math":"\\therefore \\int_{-\\infty}^{\\infty}\\frac{dx}{(x^2+1)^3} = \\frac{3\\pi}{8}"}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-partc-q4","number":4,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Establish the following result for orthogonality of Legendre Polynomial\n\n$$\\int_{-1}^1 P_m(x) P_n(x) dx = 0 \\text{, if } m \\ne n.$$"}],"answer":{"summary":"Orthogonality of Legendre polynomials (integral of Pm*Pn from -1 to 1 = 0 for m != n) is established by showing Pm and Pn satisfy Legendre's equation and using Green's identity (Sturm-Liouville theory).","explanation":[{"type":"paragraph","heading":"Legendre's Differential Equation","paragraph":"Pn(x) satisfies:"},{"type":"math","math":"\\frac{d}{dx}\\left[(1-x^2)\\frac{dP_n}{dx}\\right] + n(n+1)P_n = 0 \\quad \\cdots (1)"},{"type":"math","math":"\\frac{d}{dx}\\left[(1-x^2)\\frac{dP_m}{dx}\\right] + m(m+1)P_m = 0 \\quad \\cdots (2)"},{"type":"paragraph","heading":"Multiply and Integrate","paragraph":"Multiply (1) by Pm and (2) by Pn, then subtract:"},{"type":"math","math":"P_m\\frac{d}{dx}\\left[(1-x^2)P'_n\\right] - P_n\\frac{d}{dx}\\left[(1-x^2)P'_m\\right] = [m(m+1)-n(n+1)]P_m P_n"},{"type":"paragraph","paragraph":"The left side can be written as:"},{"type":"math","math":"\\frac{d}{dx}\\left[(1-x^2)(P_m P'_n - P_n P'_m)\\right]"},{"type":"paragraph","heading":"Integrate from -1 to 1","paragraph":"Integrating over [-1, 1]:"},{"type":"math","math":"\\left[(1-x^2)(P_m P'_n - P_n P'_m)\\right]_{-1}^{1} = [m(m+1)-n(n+1)]\\int_{-1}^{1}P_m P_n\\,dx"},{"type":"paragraph","heading":"Evaluate Boundary Term","paragraph":"At x = 1 and x = -1, (1 - x^2) = 0. Therefore the left side is 0."},{"type":"math","math":"0 = [m(m+1) - n(n+1)]\\int_{-1}^{1}P_m(x)P_n(x)\\,dx"},{"type":"paragraph","heading":"Conclude Orthogonality","paragraph":"When m != n, m(m+1) != n(n+1), so the factor [m(m+1) - n(n+1)] != 0. Therefore:"},{"type":"math","math":"\\int_{-1}^{1}P_m(x)P_n(x)\\,dx = 0, \\quad m \\ne n"},{"type":"paragraph","paragraph":"This is the orthogonality property of Legendre polynomials."}]}},{"id":"rtu-ee-ec-eee-s4-advanced-engineering-mathematics-2-2024-partc-q5","number":5,"marks":10,"tags":["Part C"],"text":[{"type":"paragraph","content":"Apply the Gram-Schmidt process to the vectors $\\beta_1 = (1, 0, 1)$, $\\beta_2 = (1, 0, -1)$, and $\\beta_3 = (0, 3, 4)$ to obtain an orthonormal basis for $V_3(R)$ with the standard inner product."}],"answer":{"summary":"Applying the Gram-Schmidt process to beta1=(1,0,1), beta2=(1,0,-1), beta3=(0,3,4) gives the orthonormal basis: e1=(1/sqrt(2),0,1/sqrt(2)), e2=(1/sqrt(2),0,-1/sqrt(2)), e3=(0,1,0).","explanation":[{"type":"paragraph","heading":"Step 1: Orthogonalize beta1","paragraph":"Set alpha1 = beta1 = (1, 0, 1)."},{"type":"math","math":"\\|\\alpha_1\\|^2 = 1^2 + 0^2 + 1^2 = 2"},{"type":"paragraph","heading":"Step 2: Orthogonalize beta2","paragraph":"Compute projection of beta2 onto alpha1:"},{"type":"math","math":"\\langle \\beta_2, \\alpha_1 \\rangle = (1)(1)+(0)(0)+(-1)(1) = 0"},{"type":"math","math":"\\alpha_2 = \\beta_2 - \\frac{\\langle\\beta_2,\\alpha_1\\rangle}{\\|\\alpha_1\\|^2}\\alpha_1 = (1,0,-1) - 0\\cdot(1,0,1) = (1,0,-1)"},{"type":"math","math":"\\|\\alpha_2\\|^2 = 1 + 0 + 1 = 2"},{"type":"paragraph","heading":"Step 3: Orthogonalize beta3","paragraph":"Compute projections of beta3 = (0,3,4) onto alpha1 and alpha2:"},{"type":"math","math":"\\langle \\beta_3, \\alpha_1 \\rangle = 0\\cdot1+3\\cdot0+4\\cdot1 = 4"},{"type":"math","math":"\\langle \\beta_3, \\alpha_2 \\rangle = 0\\cdot1+3\\cdot0+4\\cdot(-1) = -4"},{"type":"math","math":"\\alpha_3 = \\beta_3 - \\frac{4}{2}\\alpha_1 - \\frac{-4}{2}\\alpha_2 = (0,3,4) - 2(1,0,1) + 2(1,0,-1)"},{"type":"math","math":"= (0,3,4) - (2,0,2) + (2,0,-2) = (0,3,0)"},{"type":"math","math":"\\|\\alpha_3\\|^2 = 9 \\implies \\|\\alpha_3\\| = 3"},{"type":"paragraph","heading":"Step 4: Normalize","paragraph":"Divide each vector by its norm:"},{"type":"math","math":"e_1 = \\frac{\\alpha_1}{\\|\\alpha_1\\|} = \\frac{1}{\\sqrt{2}}(1,0,1) = \\left(\\frac{1}{\\sqrt{2}},\\ 0,\\ \\frac{1}{\\sqrt{2}}\\right)"},{"type":"math","math":"e_2 = \\frac{\\alpha_2}{\\|\\alpha_2\\|} = \\frac{1}{\\sqrt{2}}(1,0,-1) = \\left(\\frac{1}{\\sqrt{2}},\\ 0,\\ -\\frac{1}{\\sqrt{2}}\\right)"},{"type":"math","math":"e_3 = \\frac{\\alpha_3}{\\|\\alpha_3\\|} = \\frac{1}{3}(0,3,0) = (0,1,0)"},{"type":"paragraph","paragraph":"The orthonormal basis for V3(R) is {e1, e2, e3} as given above."}]}}]}],"loading":false,"error":null}}}

Q10Advanced Engineering Mathematics II

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