(Mathematics 5)
Syllabus of mathematics 5
Complex variables
Functions of complex variables, Derivatives and integrals of complex
functions, Cauchy's theorems, Taylor and Laurent’s expansions, residues and
its applications to real integrals.
Algebra
Solution of system of linear algebraic equations homogeneous and non-
homogeneous, Characteristic values and characteristic vectors, diagonalizing
matrices, Numerical solution of system of non-linear equations.
Functions of complex variables
Theory of functions of a complex
variable, is the branch of mathematical
analysis that investigates functions of
complex numbers. It is helpful in many
branches of mathematics, including
algebraic geometry, number theory, analytic
combinatorics, applied mathematics; as well
as in physics, including the branches of
hydrodynamics, thermodynamics, and
particularly quantum mechanics. By
extension, use of complex analysis also has
applications in engineering fields such as
nuclear, aerospace, mechanical and
electrical engineering.
Functions of complex variables
Equations without real solutions, such as 𝑥 2 = −1 or 𝑥 2 − 10𝑥 + 40 = 0 were observed
early in history and led to the introduction of complex numbers. By definition, a complex
number 𝒛 is an ordered pair (𝑥, 𝑦) of real numbers 𝑥 and y, written 𝑧 (𝑥, 𝑦). x is called the
real part and y the imaginary part of z, written
𝒛 = 𝒙 + 𝒊𝒚, 𝒛 = (𝒙, 𝒚), or 𝒙 = 𝑹𝒆 𝒛, 𝒚 = 𝑰𝒎 𝒛
One can see
Where 𝑖 = −1
𝑖 = −1
Addition, Multiplication 𝑖 2 = −1
𝑖 3 = −𝑖
Let 𝒛𝟏 = 𝒙𝟏 + 𝒊 𝒚𝟏 and 𝒛𝟐 = 𝒙𝟐 + 𝒊 𝒚𝟐 then
𝑖4 = 1
𝑧1 ± 𝑧2 = (𝒙𝟏 ± 𝒙𝟐 ) + 𝒊 (𝒚𝟏 ±𝒊𝒚𝟐 )
𝑖 5 = −1, …
𝑧1 . 𝑧2 = 𝒙𝟏 . 𝒙𝟐 − 𝒚𝟏 𝒚𝟐 + 𝒊 (𝒙𝟏 𝒚𝟐 +𝒙𝟐 𝒚𝟏 )
Functions of complex variables
Functions of complex variables
Complex Conjugate Numbers
Functions of complex variables
Functions of complex variables
Division of complex numbers
Functions of complex variables
Functions of complex variables
Functions of complex variables
Functions of complex variables
Functions of complex variables
Functions of complex variables
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