Core pure 2

What is Euler's relation?

e^{iθ} = cosθ + isinθ

z = re^{iθ}

sinθ and cosθ equal?

sinθ = (e^{iθ} − e^{-iθ}) / 2i

cosθ = (e^{iθ} + e^{-iθ}) / 2

What is De Moivre's theorem?

(cosθ + isinθ)^n = cos(nθ) + isin(nθ)

What are the rules for multiplying and dividing complex numbers in exponential form?

z₁z₂ = r₁r₂e^{i(θ₁+θ₂)}

z₁/z₂ = (r₁/r₂)e^{i(θ₁−θ₂)}

arg(z₁z₂) = arg(z₁) + arg(z₂)

What are the results for z^n + z^{-n} and z^n − z^{-n}?

z^n + z^{-n} = 2cos(nθ)

z^n − z^{-n} = 2isin(nθ)

Process for cos^n(θ) or sin^n(θ) ?

1. Use z + z^{-1} = 2cosθ or z − z^{-1} = 2isinθ

2. Raise to power n

3. Binomially expand LHS

4. Group terms using z^k + z^{-k} = 2cos(kθ)

5. Equate real/imaginary parts

Process for cos(nθ) or sin(nθ) as powers of sinθ/cosθ?

1. Write (cosθ + isinθ)^n using De Moivre's

2. Binomially expand RHS

3. Collect real parts → cos(nθ), imaginary parts → sin(nθ)

How do you find the sum of a complex series?

1. Convert to exponential form

2. Apply sum of geometric series formula

3. Factorise denominator

4. Extract real or imaginary part

How do you find the nth roots of a complex number w?

1. Write w in modulus-argument form and z in mod arg form too

2. Apply de Moivres, Compare r for r value, add integer mutiples to RHS (θ+2kπ)

3. Sub k = 0, 1, 2, …, n−1 for distinct roots

4. Write out complex numbers => roots form a regular n-gon on an Argand diagram

What is an nth root of unity and how do you find them?

Solutions to z^n = 1

nth root of unity: ω = e^{2πi/n}

All roots: 1, ω, ω², …, ω^{n−1}

Multiply any known root by ω to get the next

What is the method of differences and when does it apply?

If u_r = f(r) − f(r+1), then Σu_r = f(1) − f(n+1)

Write out first and last few terms, cancel the middle, add what remains

What are the steps for method of differences?

1. Express general term as f(r) − f(r+1) (often via partial fractions)

2. Write out r = 1, 2, 3, …, n−1, n vertically

3. Identify what cancels

4. Add remaining terms and simplify

What is the Maclaurin series formula?

f(x) = f(0) + f'(0)x + f''(0)x²/2! + f'''(0)x³/3! + …

Valid only if all derivatives at x=0 are finite and the series converges

What are the standard Maclaurin series for eˣ, sin x, cosx, ln(1+x), (1+x)^n

eˣ = 1 + x + x²/2! + x³/3! + … (all x)

sinx = x − x³/3! + x⁵/5! − … (all x)

cosx = 1 − x²/2! + x⁴/4! − … (all x)

ln(1+x) = x − x²/2 + x³/3 − … (−1 < x ≤ 1)

(1+x)^n = 1 + nx + n(n−1)x²/2! + … (all x)

How do you find the Maclaurin series of a function from scratch?

1. Differentiate f(x) repeatedly until a pattern emerges

2. Sub x=0 into each

3. Sub into Maclaurin formula

4. Identify pattern for general term if needed

How do you find the Maclaurin series of a compound function f(g(x))?

1. Expand f(u) in its standard series

2. Identify zeros of f(u)

3. Replace u with g(x)

4. Expand and collect up to required power

What is an integral improper and how do you evaluate it?

Improper if: one/both limits are ±∞, or f(x) undefined in [a,b]

Replace problem limit with t, evaluate, then take the limit as t → ∞ or t → problem value

Convergent if limit exists; divergent if not

How do you handle an improper integral where f(x) is undefined inside the interval?

Split at the undefined point k: ∫ₐᵇ = ∫ₐᵏ + ∫ᵏᵇ

Both must converge for original to converge

What is the mean value of f(x) over [a,b]?

Mean = 1/(b−a) · ∫ₐᵇ f(x) dx

What are the inverse trig integration results? (arctan, arcsin,ln)

∫ 1/(a²+x²) dx = (1/a)arctan(x/a) + c

∫ 1/√(a²−x²) dx = arcsin(x/a) + c, |x|<a

∫ 1/(x²−a²) dx = (1/2a)ln|(x−a)/(x+a)| + c

What are the derivatives of inverse trig functions? (arsin, arccos, arctan)

d/dx(arcsinx) = 1/√(1−x²)

d/dx(arccosx) = −1/√(1−x²)

d/dx(arctanx) = 1/(1+x²)

How do you integrate functions with a quadratic denominator?

If numerator is constant → use arctan or arcsin form

If numerator is polynomial → partial fractions

Complete the square if needed

For a²+x²: use x = a tanθ substitution

For a²−x²: use x = a sinθ substitution

How do you set up partial fractions with a quadratic factor?

One quadratic factor: (ax²+b) → (Ax+B)/(ax²+b)

Repeated quadratic: include (Ax+B)/(ax²+b) + (Cx+D)/(ax²+b)²

Numerator must be one degree lower than denominator

What is the formula for volume of revolution about the x-axis and y-axis?

V = π ∫ₐᵇ y² dx

V = π ∫ₐᵇ x² dy

What is the formula for volume of revolution about the x-axis and y-axis parametrically?

For parametric x=f(t), y=g(t): V = π ∫ y² (dx/dt) dt

For parametric: V = π ∫ x² (dy/dt) dt

What are the conversion formulas between polar and Cartesian?

x = rcosθ

y = rsinθ

r² = x² + y²

tanθ = y/x

What are the polar equations for a circle, half line, cardoid, spiral, dimple

r = a → circle, centre O, radius a

θ = α → half-line from O at angle α

r = aθ → spiral from O

r = a(p + qcosθ), p,q > 0: p=q → cardioid, p>q → dimple, p<q → inner loop

r = acos(nθ) or asin(nθ) → rose with n petals (n odd) or 2n petals (n even)

What is the polar area formula and how do you find limits?

Area = ½ ∫ₐᵝ r² dθ

Single loop: set r=0, solve for θ, take two consecutive solutions

Area between two curves: use intersection points as limits

How do you find tangents parallel or perpendicular to the initial line?

Parallel (horizontal): set dy/dθ = 0

Perpendicular (vertical): set dx/dθ = 0

Use y = rsinθ and x = rcosθ, then differentiate with respect to θ

How do you convert a Cartesian equation to polar form?

1. Replace x with rcosθ and y with rsinθ

2. Use r² = x²+y²

3. Simplify to get r = f(θ)

What are the definitions of sinh, cosh and tanh?

sinhx = (eˣ − e⁻ˣ)/2

coshx = (eˣ + e⁻ˣ)/2

tanhx = sinhx/coshx = (eˣ − e⁻ˣ)/(eˣ + e⁻ˣ)

What are the key hyperbolic identities?

cosh²x − sinh²x = 1

sinh(A±B) = sinhAcoshB ± coshAsinhB

cosh(A±B) = coshAcoshB ± sinhAsinhB

sinh2A = 2sinhAcoshA

cosh2A = 2cosh²A − 1 = 1 + 2sinh²A

What is Osborn's rule?

Replace cosA → coshA and sinA → sinhA, BUT any product of two sin terms becomes minus the product of two sinh term

e.g. sin²A → −sinh²A, sinAsinB → −sinhAsinhB

What are the inverse hyperbolic functions in log form? (arsinh, arcosh, artanh)

arsinhx = ln(x + √(x²+1))

arcoshx = ln(x + √(x²−1)), x≥1

artanhx = ½ln((1+x)/(1−x)), |x|<1

What are the derivatives of hyperbolic and inverse hyperbolic functions?

d/dx(sinhx) = coshx

d/dx(coshx) = sinhx

d/dx(tanhx) = sech²x

d/dx(arsinhx) = 1/√(x²+1)

d/dx(arcoshx) = 1/√(x²−1), x>1 | d/dx(artanhx) = 1/(1−x²), |x|<1

What are the key hyperbolic integrals? (sinhx, coshx, tanhx, 1/√(x²+a²) , 1/√(x²−a²))

∫sinhx dx = coshx + c

∫coshx dx = sinhx + c

∫tanhx dx = ln(coshx) + c

∫ 1/√(x²+a²) dx = arsinh(x/a) + c

∫ 1/√(x²−a²) dx = arcosh(x/a) + c, x>a

What substitutions are used for hyperbolic integration?

√(x²+a²) → use x = a sinhu

√(x²−a²) → use x = a coshu

For completing the square on quadratic denominators: let u = (linear part), then match to standard form

How do you solve a first-order DE using the integrating factor method?

1. Rearrange to dy/dx + P(x)y = Q(x)

2. I.F. = e^{∫P(x)dx}

3. Multiply through by I.F.

4. 4. LHS becomes d/dx(y · I.F.)

5. 5. Integrate both sides | 6. Solve for y

What are the three cases for the auxiliary equation of a 2nd-order homogeneous DE: ay'' + by' + cy = 0?

b²>4ac: two real roots α,β → y = Ae^{αx} + Be^{βx}

b²=4ac: repeated root α → y = (A+Bx)e^{αx}

b²<4ac: complex roots p±qi → y = e^{px}(Acosqx + Bsinqx)

How do you find the particular integral (PI) for a non-homogeneous 2nd-order DE?

Match form of f(x): constant → λ

linear → λ+μx

quadratic → λ+μx+νx²

e^{kx} → λe^{kx} | cos/sin(ωx) → λcos(ωx)+μsin(ωx)

Substitute PI into DE to find coefficients

If PI same form as CF: multiply PI by x

What is the general solution to a 2nd-order non-homogeneous DE?

General solution = Complementary Function (CF) + Particular Integral (PI)

CF solves ay''+by'+cy=0

PI satisfies the full equation

Use boundary conditions to find A and B

How do you solve coupled first-order DEs?

1. Rearrange one equation to isolate one variable

2. 2. Differentiate

3. 3. Substitute into the other equation to form a 2nd-order DE in one variable

4. 4. Solve as usual

5. 5. Back-substitute to find the other variable

What is the differential equation for simple harmonic motion (SHM)?

ẍ = −ω²x

What are the standard SHM solutions depending on initial conditions?

If x=0 at t=0: x = asin(ωt)

If x=a at t=0 (starts at max): x = acos(ωt)

General: x = Rsin(ωt+α)

What is damped harmonic motion and what are the three cases?

DE: ẍ + kẋ + ω²x = 0

k²>4ω²: heavy damping → two real roots, no oscillations

k²=4ω²: critical damping → repeated root, no oscillations

k²<4ω²: light damping → complex roots, oscillations decay exponentially

What is forced harmonic motion?

ẍ + kẋ + ω²x = f(t)

Non-homogeneous 2nd-order DE

Solve CF (homogeneous) + find PI based on form of f(t)

General solution = CF + PI

What are the steps for modelling flow in/out problems?

1. Find amount of substance after time t

2. Express concentration as amount/volume

3. Rate in = (flow rate in) × (concentration in)

4. Rate out = (flow rate out) × (current concentration)

5. Form DE: d(amount)/dt = rate in − rate out

6. Solve using integrating factor

How to find the period?

Period = 2π/ω

How to find the amplitude?

Amplitude = maximum displacement

v = 0 at max displacement; v = max at x = 0