A complex number is expressed as z = a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit with the property i² = -1.
Example:
z = 3 + 4i → Re(z) = 3, Im(z) = 4
Operations include addition, subtraction, multiplication, and division, treating i² = -1.
Example:
(2 + 3i) + (1 - 2i) = 3 + i
(2 + 3i) × (1 - 2i) = 8 - i
The polar form of a complex number is z = r(cosθ + i sinθ), where r = |z| and θ = arg(z).
Example:
z = 1 + i → r = √2, θ = π/4
z = √2 (cos π/4 + i sin π/4)
De Moivre’s Theorem: (cosθ + i sinθ)^n = cos(nθ) + i sin(nθ)
Example:
z = 1 + i = √2(cos π/4 + i sin π/4)
z² = 2i (using De Moivre’s Theorem)
Functions like f(z) = z² take complex inputs and return complex outputs.
Example:
f(1 + i) = (1 + i)² = 2i
Elementary functions include exponential, trigonometric, and logarithmic functions extended to complex domain.
Example:
e^{iθ} = cos θ + i sin θ (Euler’s Formula)
Inverse trigonometric functions in complex form involve logarithmic expressions.
Example:
cos⁻¹(z) = -i ln(z + √(z² - 1))
A sequence is an ordered list of numbers, which can be finite or infinite.
Example:
Finite: 1, 3, 5, 7
Infinite: 1, 1/2, 1/3, 1/4, ...
A sequence is monotonic if it is either non-increasing or non-decreasing. A sequence is bounded if it has an upper and/or lower limit.
Example:
Monotonic Increasing: 1, 2, 3, 4, ...
Bounded: aₙ = (-1)ⁿ, which lies between -1 and 1
If a sequence approaches a specific value as n → ∞, it is said to converge to that value.
Example:
lim (1/n) = 0 as n → ∞ → Convergent sequence
A series is the sum of terms of a sequence. The nth partial sum is the sum of the first n terms.
Example:
Series: 1 + 1/2 + 1/4 + 1/8 + ...
Partial sum S₄ = 1 + 1/2 + 1/4 + 1/8 = 1.875
A series is convergent if its sequence of partial sums approaches a limit. Theorems help determine convergence.
Example:
Harmonic series: ∑1/n diverges
Geometric series: ∑(1/2)ⁿ converges
An alternating series changes signs and may converge even if the absolute series diverges.
Example:
Alternating harmonic series: 1 - 1/2 + 1/3 - 1/4 + ... → convergent (conditionally)
Several tests determine whether a series converges or diverges:
Example (Ratio Test):
aₙ = 1/n!
aₙ₊₁/aₙ = 1/(n+1)! × n! = 1/(n+1) → 0 as n → ∞ ⇒ Converges
Vector differentiation involves differentiating each component of a vector function with respect to a variable.
Example:
If ⇡(t) = t i + t² j + e^t k,
then d⇡/dt = i + 2t j + e^t k
A scalar field assigns a scalar value to every point in space, while a vector field assigns a vector.
Example:
Scalar field: φ(x, y, z) = x² + y² + z²
Vector field: F(x, y, z) = x i + y j + z k
The gradient of a scalar field is a vector that points in the direction of the greatest rate of increase of the function.
Example:
∇φ = ∇(x² + y² + z²) = 2x i + 2y j + 2z k
The directional derivative of a scalar field φ in the direction of a unit vector u gives the rate of change of φ in that direction.
Formula:
D_u φ = ∇φ · u
Example:
If φ = x² + y² and u = (1/√2)(i + j), then D_u φ = 2x/√2 + 2y/√2 = √2(x + y)
Divergence measures the net rate of flow out of a point in a vector field. It is a scalar.
Example:
F = xi + yj + zk ⇒ div F = ∂x/∂x + ∂y/∂y + ∂z/∂z = 3
The curl of a vector field measures its tendency to rotate about a point. It is a vector.
Example:
F = -y i + x j ⇒ curl F = ∇ × F = 0 i + 0 j + 2 k
Gradient: Direction of steepest ascent.
Divergence: Net outflow per unit volume (e.g., fluid spreading from a point).
Curl: Rotational tendency (e.g., swirling fluid).
A function f(x) is periodic if there exists a constant T such that f(x + T) = f(x) for all x. The smallest such T is called the period.
Example:
f(x) = sin(x), cos(x) are periodic with period 2π
The Fourier series of a periodic function f(x) with period 2π is given by:
f(x) = a₀/2 + Σ (aₙ cos(nx) + bₙ sin(nx)), n = 1 to ∞
where:
aₙ = (1/π) ∫_{-π}^{π} f(x) cos(nx) dx
bₙ = (1/π) ∫_{-π}^{π} f(x) sin(nx) dx
- For even functions: bₙ = 0 → Only cosine terms remain.
- For odd functions: aₙ = 0 → Only sine terms remain.
Example:
f(x) = x on [-π, π] is odd → Fourier sine series only
Defined for functions on [0, L] extended as even or odd functions to use sine or cosine series.
Example:
For f(x) = x on [0, π], use sine series:
f(x) = Σ bₙ sin(nπx/π)
In this method, the equation can be written in the form f(y) dy = g(x) dx and then integrated on both sides.
Example:
(dy/dx) = xy → dy/y = x dx
Integrate: ln|y| = x²/2 + C
A first-order differential equation is called homogeneous if it can be expressed as dy/dx = F(y/x). Use substitution y = vx (or x = vy).
Example:
(dy/dx) = (x + y)/(x - y)
Let y = vx ⇒ dy/dx = v + x dv/dx
Substitute and solve in terms of v and x.
An equation M(x, y) dx + N(x, y) dy = 0 is exact if ∂M/∂y = ∂N/∂x. Solution exists as a function φ(x, y) = C.
Example:
(2xy + y²) dx + (x² + 2xy) dy = 0
Check exactness: ∂M/∂y = ∂N/∂x = 2x + 2y
The general form is dy/dx + P(x)y = Q(x). It is solved using an integrating factor IF = e^(∫P(x) dx).
Example:
dy/dx + y = e^x
IF = e^x → Multiply through and integrate:
e^x y = ∫e^x × e^x dx = ∫e^(2x) dx
⇒ y = (1/2)e^x + C/e^x
The form is dy/dx + P(x)y = Q(x)yⁿ. Divide by yⁿ and substitute z = y^(1-n) to reduce it to linear form.
Example:
dy/dx + y = y² → Divide by y²:
y⁻² dy/dx + y⁻¹ = 1
Let z = y⁻¹ ⇒ dz/dx = -y⁻² dy/dx
Use substitution and solve.
This method is applied to linear equations of first order and degree using the IF = e^(∫P(x) dx).
Example:
dy/dx + 2y = 3x
IF = e^(2x), Multiply:
e^(2x) dy/dx + 2e^(2x)y = 3x e^(2x)
Integrate both sides and solve for y.
These are of the form: a d²y/dx² + b dy/dx + cy = 0. The auxiliary equation helps find the general solution.
Example:
d²y/dx² - 5 dy/dx + 6y = 0
Auxiliary Eq: m² - 5m + 6 = 0 → m = 2, 3
Solution: y = C₁e^{2x} + C₂e^{3x}
If auxiliary equation has complex or repeated roots, the solution is adjusted accordingly:
Case 1 (Complex Roots):
m = α ± iβ ⇒ y = e^{αx}(C₁ cos βx + C₂ sin βx)
Case 2 (Repeated Roots):
m = r (double root) ⇒ y = (C₁ + C₂x)e^{rx}
Uses operator D where D = d/dx. The equation is written using D for simplification.
Example:
(D² - 4D + 4)y = 0 → (D - 2)² y = 0
Solution: y = (C₁ + C₂x)e^{2x}
If f(x) is of standard form (e^ax, sin(ax), cos(ax), xⁿ), PI can be found directly.
Example:
(D - 2)y = e^{2x} → PI = e^{2x}/(D - 2) = x e^{2x}
Used for non-homogeneous equations with standard right-hand terms. Assume a form of solution and determine constants.
Example:
d²y/dx² - y = x
Try y_p = Ax + B, plug into equation, solve for A, B
Equations of the form: x² d²y/dx² + a x dy/dx + b y = 0. Change variable x = e^t to convert to constant coefficient equation.
Example:
x² d²y/dx² - x dy/dx + y = 0 → Change x = e^t
Apply inverse operator to RHS function using known identities or shortcuts for exponential, trigonometric, and polynomial forms.
Example:
(D² - 4)y = sin(2x) → PI = sin(2x)/(D² + 4)