Mathematical and physical tools for R&I engineers

This course is a refresher course in mathematics, with applications to physics in the context of communication systems.

Course outline:

Math part
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Complex

    introduction to the complex plane.
Real axis imaginary axis.
Cartesian coordinates
Polar coordinates
sum of two complex numbers
product of two complex numbers
product in Cartesian coordinates
product in polar coordinates
Polar coordinates, use in physics
Moivre formula and transition to Euler
Moivre formula
Euler formulae
Hyperbolic trigonometry

Function of a real variable

    Definition of a function
Definition domain
Limits
Limit at a point,
Limit at infinity,
horizontal asymptote,
oblique asymptote
Calculation of limits
Limit at a point
Limit at infinity
Comparison of growth, ln, polynomials, exponential
Continuities
Definition
Derivatives
Variations
Definition of derivative
Differential
Calculation of derivatives
Derivatives of common functions
Sum
product
composite function
power
Series developments, limited developments, L'Hospital's rule
Development in integer series
Limited developments
L'Hospital's rule
Table of variations
Graph
Possible reciprocal functions, injective and bijective functions.
Notion of reciprocal function
Injective function
Surjective function
Most common functions

Integral calculation

    Riemann integral
Definition:
Primitive
Bounded integral
Properties
Additivity of functions
Product by a constant
Chasles relation
Mean theorem:
Definition of the mean:
Mean theorem :
Immediate primitives
Decomposition into sums, linearization, compound functions
Integration by parts
Principle of integration by parts
Area of application of integration by parts
Bijective change of variable
Case of an indefinite integral
in practice two cases:
Primitives of rational fractions
Types of functions
Decomposition into simple elements
Euclidean division
Finding the poles

differential equations

    General considerations
First-order differential equations
Shape of function
a) solve equation without second member
b) find particular solution
b1 obvious solution
b2 polynomial second member
b3 sine or cosine second member
b4 exponential second member
b5 variation of constant
Second-order differential equations of the second order
Allure of the function
a) solving the equation without a second member
Real roots
Complex conjugate roots
b) finding the particular solution
b1 obvious solution
b2 polynomial second member
b3 sine or cosine second member
b4 exponential second member
b5 variation of the constant
Fourier series

    Definitions
Writing in complexes
Notion of convergence Dirichlet's theorem
Gibbs phenomenon
Shannon's theorem
Parseval's theorem 

Fourier transform

    From Fourier series to Fourier transform

Physical part
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Dipole
Lorentz force
Voltage
Current
Transient dipole
Resistive dipole
Inductive dipole
Capacitive dipole
Sinusoidal dipole
Resistive dipole
Inductive dipole
Capacitive dipole

Electrical networks. Association of dipoles

    Association in series
Mesh law
RLC circuit
Resonance frequency
Association in parallel
Node law
LC circuit
Plug frequency