Complex Analysis
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Transcript of Complex Analysis
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Complex Analysis
Prepared by
Dr. Taha MAhdy
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Complex analysis importance
• Complex analysis has not only transformed the world of mathematics, but surprisingly, we find its application in many areas of physics and engineering.
• For example, we can use complex numbers to describe the behavior of the electromagnetic field.
• In atomic systems, which are described by quantum mechanics, complex numbers and complex functions play a central role,
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What is a complex number
• It is a solution for the equation
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The Algebra of Complex Numbers
• More general complex numbers can be written down. In fact, using real numbers a and b we can form a complex number:
c = a + ib
• We call a the real part of the complex number c and refer to b as the imaginary part of c.
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Addition , subtraction, multiplication
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Complex conjugate
• The complex conjugate is:
• Note that
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Complex conjugate
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Division is defiened in terms of conjugate of the denominator
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Graphical representation of complex number
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Complex Variables
• A Complex Variable can assume any complex value
• We use z to represent a complex variable.
z = x + jy
• We can graph complex numbers in the x-y plane, which we sometimes call the complex plane or the z plane.
• We also keep track of the angle θ that this vector makes with the real axis.
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Very Important complex transformations
It appears that complex numbers are not so “imaginary” after all;
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The Polar Representation
• Let z = x + iy is the Cartesian representation of a complex number.
• To write down the polar representation, we begin with the definition of the polar coordinates (r,θ ):
x = r cosθ ; y = r sinθ
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The Polar Representation
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The Polar Representation
• Note that r > 0 and that we have
• tanθ = y / x as a means to convert between polar and Cartesian representations.
• The value of θ for a given complex number is called the argument of z or arg z.
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THE ARGUMENT OF Z
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EULER’S FORMULA
• Euler’s formula allows us to write the expression cosθ + i sinθ in terms of a complex exponential.
• This is easy to see using a Taylor series expansion.
• First let’s write out a few terms in the well-known Taylor expansions of the trigonometric functions cos and sin:
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Note the similarity
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EULER’S FORMULA
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EULER’S FORM
• These relationships allow us to write a complex number in complex exponential form or more commonly polar form. This is given by
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EULER’S FORM operations
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EULER’S FORM operations
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EULER’S FORM operations
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DE MOIVRE’S THEOREM
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Assignment
• Solve the problems of the chapter