series expansions and complex numbers

Question 1

what is the real and imaginary part of a complex number z = a + ib

Answer 1

a = real
ib = imaginary

Question 2

conjugation: z = a + ib, z* =

Answer 2

z* = a - ib

Question 3

i =

Answer 3

i = sqrt(-1)

Question 4

i^2 =

Answer 4

i^2 = -1

Question 5

argand diagram: r^2 =

Answer 5

r^2 = x^2 + y^2

Question 6

argand diagram: tan(theta) =

Answer 6

tan(theta) = y/x

Question 7

polar coordinates to cartesian coordinates

Answer 7

z = r(cos(theta) + isin(theta))

Question 8

r =

Answer 8

r = radius (vector length)

Question 9

theta =

Answer 9

theta = angle

Question 10

omega =

Answer 10

omega = angular frequency

Question 11

if omega is constant, phi (t) =

Answer 11

phi(t) = omega*t + theta, with theta = phi at t = 0

Question 12

formula for locus of point on the unit circle with theta = 0, z =

Answer 12

z = cos (omegat) + isin(omegat)

Question 13

euler’s formula, exp(ix) = and exp(-ix) =

Answer 13

exp(ix) = cos(x)+ isin(x) and exp(-ix) = cos(x) - isin(x)

Question 14

euler’s formula, cos(x) =

Answer 14

cos(x) = (exp(ix) + exp(-ix))/2

Question 15

euler’s formula, sin(x) =

Answer 15

sin(x) = i(exp(-ix) - exp(ix))/2

Question 16

de moivre theorem, (cos(theta) + isin(theta))^n=

Answer 16

(cos(theta) + isin(theta))^n = exp(intheta) = (cos(ntheta) + isin(n*theta))

Question 17

circular motion, z = cos (omegat) + isin(omegat) =

Answer 17

z = cos (omegat) + isin(omegat) = exp(iomegat)

Question 18

flip angle in mri, phi =

Answer 18

phi = omega1tp = gammaB1*tp, with tp = pulse duration