Podcast
Questions and Answers
What is the correct expression for the complex function w when z = 1 + 3i?
What is the correct expression for the complex function w when z = 1 + 3i?
- -8 + 9i (correct)
- 2 + 5i
- 2 + 6i
- 1 + 6i
The functions that are differentiable at some point of their domain are called continuous functions.
The functions that are differentiable at some point of their domain are called continuous functions.
False (B)
What are the Cauchy-Riemann equations?
What are the Cauchy-Riemann equations?
ux = vy, uy = -vx
A complex function f(z) is said to have a limit L as z approaches to ______.
A complex function f(z) is said to have a limit L as z approaches to ______.
Match the following terms with their definitions:
Match the following terms with their definitions:
Which of the following functions is NOT analytic?
Which of the following functions is NOT analytic?
The function f(z) = z^3 is analytic because it satisfies the Cauchy-Riemann equations.
The function f(z) = z^3 is analytic because it satisfies the Cauchy-Riemann equations.
What form do the Cauchy-Riemann equations take for a function defined in terms of polar coordinates?
What form do the Cauchy-Riemann equations take for a function defined in terms of polar coordinates?
The function f(z) = ln|z| + i arg z is considered analytic because it satisfies _____.
The function f(z) = ln|z| + i arg z is considered analytic because it satisfies _____.
Match the following functions with their analytic status:
Match the following functions with their analytic status:
Which of the following pairs satisfies the Cauchy-Riemann equations in domain D?
Which of the following pairs satisfies the Cauchy-Riemann equations in domain D?
The harmonic functions u and v must always satisfy Laplace's equation in the entire complex plane.
The harmonic functions u and v must always satisfy Laplace's equation in the entire complex plane.
What is the expression for the harmonic conjugate of u = x^2 - y^2 - y?
What is the expression for the harmonic conjugate of u = x^2 - y^2 - y?
If $u = x^2 - y^2 - y$, then the condition for u to be harmonic is that __________.
If $u = x^2 - y^2 - y$, then the condition for u to be harmonic is that __________.
Match the functions with their respective properties:
Match the functions with their respective properties:
What is the imaginary unit 'i' defined as?
What is the imaginary unit 'i' defined as?
Two complex numbers are equal if their real parts are equal, regardless of their imaginary parts.
Two complex numbers are equal if their real parts are equal, regardless of their imaginary parts.
What is the polar form of a complex number represented as?
What is the polar form of a complex number represented as?
The product of the complex numbers z1 = 3 + 4i and z2 = 1 + 2i is _____
The product of the complex numbers z1 = 3 + 4i and z2 = 1 + 2i is _____
Match the complex operations to their results:
Match the complex operations to their results:
Flashcards
Complex Number
Complex Number
A complex number is an ordered pair (x, y) where x is the real part and y is the imaginary part, written as z = x + iy.
Imaginary Unit (i)
Imaginary Unit (i)
The imaginary unit i is defined as i = √(-1), used to represent the imaginary part of a complex number.
Complex Conjugate
Complex Conjugate
The complex conjugate of z = x + iy is z* = x - iy, reflecting it across the real axis.
Polar Form of Complex Number
Polar Form of Complex Number
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Algebra of Complex Numbers
Algebra of Complex Numbers
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Complex Variable
Complex Variable
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Complex Function
Complex Function
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Analytic Functions
Analytic Functions
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Cauchy-Riemann Equations
Cauchy-Riemann Equations
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Derivative of Complex Function
Derivative of Complex Function
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Verification of analyticity
Verification of analyticity
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Polar form in C.R. equations
Polar form in C.R. equations
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Function f(z) = z^3
Function f(z) = z^3
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Harmonic function
Harmonic function
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Laplace's equation
Laplace's equation
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Harmonic conjugate
Harmonic conjugate
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Study Notes
Complex Numbers
- Complex numbers are used to solve equations that do not have real solutions.
- A complex number (z) is represented as z = x + iy, where x is the real part and y is the imaginary part.
- Imaginary unit (i) is defined as i² = -1.
- Equality of two complex numbers (z₁ = x₁ + iy₁, z₂ = x₂ + iy₂) holds if and only if their real and imaginary parts are equal (x₁ = x₂ and y₁ = y₂).
Algebra of Complex Numbers
- Sum: (z₁ + z₂) = (x₁ + x₂) + i (y₁ + y₂).
- Product: (z₁z₂) = (x₁x₂ - y₁y₂) + i (x₁y₂ + x₂y₁).
- Difference: (z₁ - z₂) = (x₁ - x₂) + i (y₁ - y₂).
- Quotient: (z₁/z₂) = [(x₁x₂ + y₁y₂)/(x₂² + y₂²)] + i [(x₂y₁ - x₁y₂)/(x₂² + y₂²)].
Geometrical Representation
- Complex numbers can be represented as points in a coordinate system.
- The horizontal axis is the real axis (Re(z)) and the vertical axis is the imaginary axis (Im(z)).
- A complex number z = x + iy is plotted as the point (x, y).
- The distance from the origin to the point (x, y) represents the absolute value or modulus of the complex number |z| = √(x² + y²).
Complex Conjugate
- The complex conjugate of a complex number z = x + iy is denoted as z* = x - iy.
- Geometrically, the complex conjugate is the reflection of the complex number across the real axis.
Polar Form
- A complex number z = x + iy can be represented in polar form as z = reiθ, where r = √(x² + y²) and θ = tan⁻¹(y/x).
- r is the modulus (distance from the origin) and θ is the argument (angle with the positive real axis).
Circles and Discs
- |z - z₀| = r represents a circle centered at z₀ = x₀ + iy₀ with radius r.
- Region |z - z₀| ≤ r represents a closed circular disc.
- Region |z - z₀| > r represents the exterior of the circle.
- Region r₁ < |z - z₀| < r₂ represents an annulus (ring-shaped region) between two circles.
Half Planes
- The set of all complex numbers z=x+iy such that Re(z) > k is the right half-plane.
- The set of all complex numbers z=x+iy such that Re(z) < k is the left half-plane.
- The set of complex numbers z=x+iy such that Im(z) > k is the upper half-plane
- The set of complex numbers z=x+iy such that Im(z) < k is the lower half-plane.
Complex Functions
- A complex function f(z) maps a complex number z to another complex number w.
- The function f(z) maps points from the z-plane to the w-plane
- f(z)=z² is an example of a complex function.
Continuity and Differentiability
- A complex function is continuous at a point if the limit of the function as z approaches that point equals the value of the function at that point.
- A complex function is differentiable at a point if the derivative of the function at that point exists.
Analytic Functions/ Regular Functions
- A function f(z) is analytic in a domain if it is differentiable at every point within that domain.
- Cauchy-Riemann (CR) equations are necessary conditions for a function to be analytic.
- A function which satisfies CR equations in a domain is differentiable in that domain.
Harmonic Functions
- A function is harmonic in a domain if it satisfies Laplace's equation in the domain
Linear Fractional Transformations (LFTs/ Bilinear Transformations)
- A transformation of the form w = (az + b)/(cz + d), where a, b, c, and d are complex numbers, and ad – bc ≠ 0.
- Every LFT maps circles onto circles (or lines considered as degenerate circles)
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Description
This quiz covers the fundamentals of complex numbers, including their definition, algebraic operations, and geometrical representation. Understand key concepts like real and imaginary parts, and how to perform addition, subtraction, multiplication, and division of complex numbers.
