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Potential Energy of a Particle | Work done by Conservative Forces

Potential Energy of a Particle | Work done by Conservative Forces

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Why Vertical Velocity changes & Horizontal does not?

Why Vertical Velocity changes & Horizontal does not?

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Free Body Diagram | Blocks & Pulleys | F= ma #6

Free Body Diagram | Blocks & Pulleys | F= ma #6

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The Rocket Equation

The Rocket Equation

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 01: Analytic functions of a complex variable (Part I)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 01: Analytic functions of a complex variable (Part I)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 02: Analytic functions of a complex variable (Part II)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 02: Analytic functions of a complex variable (Part II)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 03: Calculus of residues (Part I)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 03: Calculus of residues (Part I)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 04: Calculus of residues (Part II)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 04: Calculus of residues (Part II)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 05: Calculus of residues (Part III)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 05: Calculus of residues (Part III)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 06: Calculus of residues (Part IV)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 06: Calculus of residues (Part IV)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 07: Linear response; dispersion relations (Part I)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 07: Linear response; dispersion relations (Part I)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 08: Linear response; dispersion relations (Part II)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 08: Linear response; dispersion relations (Part II)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 09: Analytic continuation and the gamma function (Part I)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 09: Analytic continuation and the gamma function (Part I)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 10: Analytic continuation and the gamma function (Part II)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 10: Analytic continuation and the gamma function (Part II)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 11: Möbius transformations (Part I)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 11: Möbius transformations (Part I)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 12: Möbius transformations (Part II)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 12: Möbius transformations (Part II)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 13: Möbius transformations (Part III)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 13: Möbius transformations (Part III)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 14: Multivalued functions; integral representations (Part I)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 14: Multivalued functions; integral representations (Part I)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 15: Multivalued functions; integral representations (Part II)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 15: Multivalued functions; integral representations (Part II)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 16: Multivalued functions; integral representations (Part III)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 16: Multivalued functions; integral representations (Part III)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 17: Multivalued functions; integral representations (Part IV)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 17: Multivalued functions; integral representations (Part IV)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 18: Laplace transforms (Part I)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 18: Laplace transforms (Part I)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 19: Laplace transforms (Part II)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 19: Laplace transforms (Part II)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 20: Fourier transforms (Part I)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 20: Fourier transforms (Part I)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 21: Fourier transforms (Part II)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 21: Fourier transforms (Part II)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 22: Fourier transforms (Part III)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 22: Fourier transforms (Part III)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 23: Fundamental Green function for Δ2(Part I)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 23: Fundamental Green function for Δ2(Part I)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 24: Fundamental Green function for Δ2 (Part II)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 24: Fundamental Green function for Δ2 (Part II)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 25: The diffusion equation (Part I)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 25: The diffusion equation (Part I)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 26: The diffusion equation (Part II)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 26: The diffusion equation (Part II)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 27: The diffusion equation (Part III)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 27: The diffusion equation (Part III)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 28: The diffusion equation (Part IV)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 28: The diffusion equation (Part IV)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 29: Green function for (Δ2 + k2); nonrelativistic scattering (Part I)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 29: Green function for (Δ2 + k2); nonrelativistic scattering (Part I)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 30: Green function for (Δ2 + k2); nonrelativistic scattering (Part II)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 30: Green function for (Δ2 + k2); nonrelativistic scattering (Part II)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 31: Green function for (Δ2 + k2); nonrelativistic scattering (Part III)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 31: Green function for (Δ2 + k2); nonrelativistic scattering (Part III)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 32: The wave equation (Part I)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 32: The wave equation (Part I)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 33: The wave equation (Part II)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 33: The wave equation (Part II)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 34: The rotation group and all that (Part I)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 34: The rotation group and all that (Part I)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 35: The rotation group and all that (Part II)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 35: The rotation group and all that (Part II)

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Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 36: The rotation group and all that (Part III)

Mathematical Physics by Prof. V. Balakrishnan (NPTEL):- Lecture 36: The rotation group and all that (Part III)

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Introduction

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Introduction

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 1 : Set, Group, Field, Ring

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 1 : Set, Group, Field, Ring

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 2 : Vector Space

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 2 : Vector Space

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 3 : Span, Linear combination of vectors

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 3 : Span, Linear combination of vectors

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 4 : Linearly dependent and independent vector, Basis

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 4 : Linearly dependent and independent vector, Basis

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 5 : Dual Space

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 5 : Dual Space

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 6 : Inner Product

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 6 : Inner Product

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 7 : Schwarz Inequality

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 7 : Schwarz Inequality

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 8 : Inner product space, Gram- Schmidt Ortho-normalization

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 8 : Inner product space, Gram- Schmidt Ortho-normalization

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 9 : Projection operator

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 9 : Projection operator

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 10: Transformation of Basis

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 10: Transformation of Basis

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 11: Transformation of Basis (Continue)

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 11: Transformation of Basis (Continue)

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 12: Unitary transformation, Similarity Transformation

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 12: Unitary transformation, Similarity Transformation

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 13: Eigen Value, Eigen Vectors

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 13: Eigen Value, Eigen Vectors

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 14: Normal Matrix

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 14: Normal Matrix

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 15: Diagonalization of a Matrix

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 15: Diagonalization of a Matrix

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 16: Hermitian Matrix

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 16: Hermitian Matrix

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 17 : Rank of a Matrix

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 17 : Rank of a Matrix

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 18 : Cayley - Hamilton Theorem, Function space

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 18 : Cayley - Hamilton Theorem, Function space

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 19: Metric Space, Linearly dependent –independent functions

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 19: Metric Space, Linearly dependent –independent functions

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 20 : Linearly dependent –independent functions (Cont), Inner Product of functions

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 20 : Linearly dependent –independent functions (Cont), Inner Product of functions

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 21: Orthogonal functions

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 21: Orthogonal functions

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 22: Delta Function, Completeness

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 22: Delta Function, Completeness

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 23: Fourier Series

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 23: Fourier Series

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 24: Fourier Series (Contd.)

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 24: Fourier Series (Contd.)

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 25: Parseval Theorem, Fourier Transform

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 25: Parseval Theorem, Fourier Transform

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 26: Parseval Relation, Convolution Theorem

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 26: Parseval Relation, Convolution Theorem

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 27: Polynomial space, Legendre Polynomial

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 27: Polynomial space, Legendre Polynomial

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 28: Monomial Basis, Factorial Basis, Legendre Basis

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 28: Monomial Basis, Factorial Basis, Legendre Basis

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 29: Complex Numbers

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 29: Complex Numbers

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 30: Geometrical interpretation of complex numbers

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 30: Geometrical interpretation of complex numbers

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 31 : de Moivre’s Theorem

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 31 : de Moivre’s Theorem

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 32 : Roots of a complex number

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 32 : Roots of a complex number

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 33 : Set of complex no, Stereographic projection

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 33 : Set of complex no, Stereographic projection

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):-Lecture 34 : Complex Function, Concept of Limit

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):-Lecture 34 : Complex Function, Concept of Limit

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 35 : Derivative of Complex Function, Cauchy-Riemann Equation

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 35 : Derivative of Complex Function, Cauchy-Riemann Equation

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 36 : Analytic Function

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 36 : Analytic Function

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 37 : Harmonic Conjugate

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 37 : Harmonic Conjugate

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 38 : Polar form of Cauchy-Riemann Equation

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 38 : Polar form of Cauchy-Riemann Equation

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 39 : Multi-valued function and Branches

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 39 : Multi-valued function and Branches

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 40 : Complex Line Integration, Contour, Regions

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 40 : Complex Line Integration, Contour, Regions

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 41: Complex Line Integration(Cont.)

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 41: Complex Line Integration(Cont.)

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):-Lecture 42: Cauchy-Goursat Theorem

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):-Lecture 42: Cauchy-Goursat Theorem

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 43: Application of Cauchy-Goursat Theorem

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 43: Application of Cauchy-Goursat Theorem

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 44: Cauchy’s Integral Formula

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 44: Cauchy’s Integral Formula

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 45: Cauchy’s Integral Formula (Contd.)

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 45: Cauchy’s Integral Formula (Contd.)

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 46: Series and Sequence

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 46: Series and Sequence

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 47: Series and Sequence (Contd.)

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 47: Series and Sequence (Contd.)

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 48:Circle and radius of convergence

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 48:Circle and radius of convergence

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 49: Taylor Series

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 49: Taylor Series

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 50: Classification of singularity

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 50: Classification of singularity

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 51: Laurent Series, Singularity

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 51: Laurent Series, Singularity

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 52: Laurent series expansion

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 52: Laurent series expansion

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 53: Laurent series expansion (Cont), Concept of Residue

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 53: Laurent series expansion (Cont), Concept of Residue

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 54: Classification of Residue

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 54: Classification of Residue

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 55: Calculation of Residue for quotient

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 55: Calculation of Residue for quotient

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 56 : Cauchy’s Residue Theorem

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 56 : Cauchy’s Residue Theorem

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):-Lecture 57 : Cauchy’s Residue Theorem (Cont)

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):-Lecture 57 : Cauchy’s Residue Theorem (Cont)

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 58 : Real Integration using Cauchy’s Residue Theorem

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 58 : Real Integration using Cauchy’s Residue Theorem

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Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 58 : Real Integration using Cauchy’s Residue Theorem (Contd. 1)

Mathematical Physics 1 by Prof. Samudra Roy (NPTEL):- Lecture 58 : Real Integration using Cauchy’s Residue Theorem (Contd. 1)

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