# Core A level Flashcards

What is the exponential form of a complex number?

z = re^iθ where

r = |z|

and θ = argz

Describe the proof for writing a complex number in exponential form

z = r(cosθ + isinθ)

Find the Maclaurin series expansion of cosθ, sinθ, and eˣ

subsitute x = iθ into the e-series, separate out the complex and real parts

you will get e^iθ = cosθ + isinθ

How do you go from the proof of exponential form of complex numbers to Euler’s identity?

e^iθ = cosθ + isinθ

θ = π

e^iθ = -1 + 0

e^iθ + 1 = 0

If z₁ = r₁e^iθ₁ and z₂ = r₂e^iθ₂, what does z₁*z₂ =

z₁*z₂ = r₁*r₂e^i(θ₁+θ₂)

What is de Moivre’s theorem?

zⁿ = rⁿ(cosnθ + isinnθ)

What can you quickly use, to prove de Moivre’s theorem for all n?

Exponential form

z + 1/z =

2cosθ

z - 1/z =

2isinθ

zⁿ + 1/zⁿ =

2cosnθ

zⁿ - 1/zⁿ =

2isin(nθ)

If zⁿ = w, what is the general solution to z, in modulus-argument form?

z = r(cos(θ+2kπ) + isin(θ+2kπ))

Describe what the roots of a complex number look like on an argand diagram

The roots lie at the vertices of a regular n-gon with its centre at the origin

n = no. of roots

When is the Maclaurin series valid?

When all the f(0), f’(0), f’‘(0), …, fʳ(0) all have finite values

When is an integral improper?

When:

one or both of the limits is infinite

f(x) is undefined at x = a, x = b, or any other point in the interval [a ,b]

If an improper integral exits, what is it described as?

If it doesn’t exist?

Exists: Convergent

Doesn’t exist: Divergent

If you have an integral, where the limits are ± ∞, how can you tell whether the integral is convergent or divergent?

Split the integral into two with limits (∞, c) and (c, -∞) where c is a number

If both integrals converge, then so does the original.

If either diverges, then the original is divergent

How do you calculate the mean value of a function? (In the interval [a, b])

= 1/(b-a) ∫ f(x) dx

where the integral limits are b and a

If the function f(x) has a mean value f-bar over the interval [a, b], and k is a real constant, then…

What is the mean value of f(x) + k?

f + k over the interval [a,b]

If the function f(x) has a mean value f-bar over the interval [a, b], and k is a real constant, then…

What is the mean value of -f(x)?

-f over the interval [a,b]

If the function f(x) has a mean value f-bar over the interval [a, b], and k is a real constant, then…

What is the mean value of kf(x)?

kf over the interval [a,b]

x = f(t)

y = g(t)

What is the volume of the solid that is generated when the parametric curve is rotated about the x-axis, between x = a and x = b, through 2π radians?

Volume = π ∫ y² dx (between x=b, and x = a)

= π ∫ y² dx/dt dt (between t=p and t=q)

where a = f(p) and b = f(q)

x = f(t)

y = g(t)

What is the volume of the solid that is generated when the parametric curve is rotated about the y-axis, between y = a and y = b, through 2π radians?

Volume = π ∫ x² dy (between y = b, and y = a)

= π ∫ x² dy/dt dt (between t = p and t = q)

where a = f(p) and b = f(q)

Polar co-ordinates:

Write x in terms of θ

rcosθ = x

Polar co-ordinates:

Write y in terms of θ

rsinθ = y

Polar co-ordinates:

Write r in terms of x and y

x² + y² = r

Polar co-ordinates:

Write θ in terms of x and y

θ = arctan (y/x)

What is the origin called in polar coordinates?

The pole

Polar co-ordinates:

What is the initial line?

Usually the positive x-axis

Polar co-ordinates:

What is the form of the coordinates? Eg, what are a and b in (a, b)

(r, θ)

Polar co-ordinates:

r = a

circle with centre O and radius a