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Cauchy's Residue Theorem

• Example 1
• Example 2

Complex variable (function of)

• Taylor Series of a function of a complex variable
• Cauchy's Residue Theorem: Example 1
• Cauchy's Residue Theorem: Example 2
• Rouche's Theorem

Differential Equations: First Order

• Solution of first order ODE using an integrating factor
• Solution of first order ODE by separation of variables

Differential Equations: Second Order

• Introduction to the Frobenius method for solving a second order linear ordinary differential equation
• Solution of a second order ordinary differential equation with constant coefficients
• Solution of Euler's Equation (a second order nonlinear ordinary differential equation)
• Solution of a second order linear ordinary differential equation using the Frobenius method

Differentiation

• Differentiation of an integral
• Logarithmic differentiation

Forces

• Resolution of forces

Integration

• ...by substitution
• Integration by substitution using hyperbolic functions


• ...contour integration
• Cauchy's Residue Theorem: Example 1
• Cauchy's Residue Theorem: Example 2


• ...hyperbolic functions
• Integration by substitution using hyperbolic functions


• ...line integrals
• Line integral (given in terms of dx, dy, dz)
• Line integral (given in terms of dr)


• ...over an area
• Calculation of surface area using integration


• ...trigonometric functions
• Integration of sin x from 0 to π/2
• Integration involving sin² x
• Integration involving sin³ x
• Integration involving tan³ x
• Integration involving products of sin x and cos x


• ...partial fractions
• Integration using partial fractions


• ...repeated integration
• Calculation of surface area using integration

Lagrange multipliers

• Lagrange multipliers for optimisation with constraints

Legendre polynomials

• Legendre polynomials and generating functions

Limits

• Evaluation of limits using L'Hospital's rule

Partial differentiation

• Partial differentiation using the chain rule

Partial fractions

• Partial fractions

Rouche's Theorem

• Rouche's Theorem

Surface area

• Calculation of surface area using integration

Taylor Series

• Taylor Series of a function of two variables
• Taylor Series of a function of a complex variable

Vectors

• Resolution of forces

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