Complex Numbers

Question 1

what is the exponential form for writing a complex number

Answer 1

z = re^iθ

Question 2

what is eulers formula

Answer 2

e^iθ = cosθ + isinθ

Question 3

what is the modulus of e^iθ

Answer 3

e^iθ = 1

Question 4

what is the modulus of e^2piin

Answer 4

1

Question 5

for 2 complex numbers z1 and z2, what does |z1*z2| equal

Answer 5

|z1*z2| = |z1| * |z2|

Question 6

what does arg(z1*z2) equal

Answer 6

arg(z1*z2) = arg(z1) + arg(z2)

Question 7

what is the effect of multiplying a complex number by i

Answer 7

rotation of pi/2 ACW

Question 8

what is the complex form of cosθ

Answer 8

cosθ = (e^iθ + e^-iθ) / 2

Question 9

what is the complex form of sinθ

Answer 9

sinθ = (e^iθ - e^-iθ) / 2i

Question 10

why does (cosθ + isinθ)^n = cosnθ + isinnθ simply speaking

Answer 10

because e^inθ = (e^iθ)^n

Question 11

what is the basis of demoivres theorem problems

Answer 11

using this idea of (cosθ + isinθ)^n = cosnθ + isinnθ to write one side in terms of the other

Question 12

what does the equation 4 + 2e^iθ tell you about the circle drawn

Answer 12

• the centre is at (4,0)

- it has a radius of 2

Question 13

how would you solve z^3 = -1

Answer 13

• write z^3 in its complex form
• add +2ipi*k to the power
• cube root both sides
• write all the solution for each k, starting from 0, until 2pi is reached

Question 14

if a = a* (a* = complex conjugate of a) what does that mean about the imaginary part of a

Answer 14

• the imaginary part of a = 0

- so a is a real number

Question 15

if p(z) = az^n + bz^n-1 + cz^n-2 +…+ qz + r, where all the coefficients are real, what does that mean for the roots of p(z)

Answer 15

the roots = 0

Question 16

what about if z1 is a root the equation

Answer 16

then the conjugate z1* is also a root

Question 17

what is this basically telling us

Answer 17

if a polynomial equation has a complex number as a root, the conjugate of that complex number will be another root

Question 18

what is the formula for coshx

Answer 18

coshx = (e^x + e^-x) / 2

Question 19

what is the formula for sinhx

Answer 19

sinhx = (e^x - e^-x) / 2

Question 20

comparing the complex formulas for cosx and coshx, what is the relationship between them

Answer 20

• cosx = coshix

- therefore cosix = coshx

Question 21

comparing the complex formulas for sinx and sinhx, what is the relationship between them

Answer 21

• i*sinx = sinhix

- therefore sinix = i*sinhx

Question 22

what is the formula for discerning ln(re^iθ)

Answer 22

ln(re^iθ) = ln|r| + ln(e^iθ) = lnr + (iθ + 2pini)

Question 23

what is the formula for the summation of a geometric progression

Answer 23

Sn = a(1 - r^n) / 1 - r

Question 24

where does i(t) = C*dv/dt come from

Answer 24

• i(t) = dQ/dt = dQ/dv * dv/dt
• dQ/dv is capacitance (C = Q/V)
• therefore i(t) = C*dv/dt

Question 25

what is the formula for current in a capacitor in terms of j,w,C and V

Answer 25

I = jwCV

Question 26

does the phase of the current in this case lag behind or lead the voltage by pi/2 rads

Answer 26

it leads

Question 27

knowing v(t) = L*dI/dt, what is the formula for the current in an inductor in terms V,j,w and L

Answer 27

I = V / jwL

Question 28

is the current through an inductor lagging or leading relative to the pd

Answer 28

its lagging

Question 29

how would you know whether the current is lagging or leading

Answer 29

• if j is in the numerator for the current calculation, its leading
• if j is in the denominator for the current calculation, its lagging

Question 30

whenever you come across 1 - j/wRC, what can you equate it to just because you can for some reason

Answer 30

re^-jθ

Question 31

what can you rewrite the int(e^-ax * cosbx) dx to in order for it to be solvable and why

Answer 31

• Re[int(e^(-ax + ibx)) dx]

- because cosnθ = Re[e^inθ]