Complex numbers

Question 1

Re(Z)

Answer 1

Z+Z*/2

Question 2

Im(Z)

Answer 2

Z-Z*/2

Question 3

Higher order derivatives of complex numbers

Answer 3

the nth derivative of a complex number in sin and cos form, multiplies the complex number by i^n

Question 4

Euler’s formula

Answer 4

e^itheta=cos(theta)+isin(theta)

Question 5

De Moivre’s theorem

Answer 5

(cos(theta)+isin(theta))^n= cos(ntheta)+isin(ntheta)

Question 6

General rule for complex number to power 1/n

Answer 6

Has n distinct roots, whose points differ by a factor 2pi/n. Connected, form an n sided regular polynomial

Question 7

general expression for complex number to power 1/n

Answer 7

cos(theta/n+2kpi/n)+isin(theta/n+2kpi/n)
where K is a rational number

Question 8

general expression for complex number to power of q, where q is a real number.
what is special about this expression

Answer 8

cos(qtheta+qk2pi)+isin(qtheta+qk2pi)

qk may not be an integer, but no solutions overlap and hence, form a circle.

Question 9

What is the ratio test for real numbers

Answer 9

For a summation to infinity, p=lim(n–>infinity) A(n+1)/A(n)
If p<1: converges
If p=1: inconclusive
If p>1: diverges

Question 10

What does the ratio test value mean for complex numbers

Answer 10

Either a diverging or converging spiral

Question 11

What is the p value for e^z and what does this mean in terms of complex numbers

Answer 11

p=0<1, therefore complex numbers with a finite magnitude have an infinite radius of convergence

Question 12

what is cos(theta) in complex form

Answer 12

e^itheta+e^-itheta/2

Question 13

what is sin(theta) in complex form

Answer 13

e^itheta-e^-itheta/2

Question 14

what is cos(Z)

Answer 14

cos(x)cosh(y)-isinxsinh(y)
[from using equations for sinh, cosh, and cos and sin in complex form]

Question 15

what is sin(Z)

Answer 15

sin(x)cosh(y)+icos(x)sinh(y)

Question 16

what is the size of cos(Z)

Answer 16

sqrt(cos^2(x)+sinh^2(y))

Question 17

what is the size of sin(Z)

Answer 17

sqrt(sin^2(x)+sinh^2(y))

Question 18

point about the size of sin(Z) and cos(Z)

Answer 18

They are unbounded

Question 19

what is the formula for the complex logarithm

Answer 19

w=lnZ=lnr+i(theta+2kpi) where argument is bounded by pi and minus pi. So in reality, k=0 to limit the complex number to one rotation

eg. W=lnZ=lnr+itheta

[NB: rules of logs hold]

principle value, value with first angle

Question 19

what is the formula for the complex logarithm

Answer 19

w=lnZ=lnr+i(theta+2kpi) where argument is bounded by pi and minus pi. So in reality, k=0 to limit the complex number to one rotation

eg. W=lnZ=lnr+itheta

[NB: rules of logs hold]

Question 20

What is the definition of continuity for a real function

Answer 20

limx–>x0 f(x)=f(x0)

Question 21

what is the definition of continuity for a complex function

Answer 21

define d=|Z-Z’|
for complex, continuous if:
limZ–>Z0 f(Z) =lim|Z-Z0|–>0 f(Z)=f(Z0)
eg. d–>0

Question 22

term for a function having a derivative at every point in a region

Answer 22

holomorphic

Question 23

Cauchy Reimann Equations

Answer 23

must satisfy BOTH necessary conditions:
* partial du/dx= partial dv/dy
* partial du/dy= partial -dv/dx

then, sufficient condition:

f’(Z)=partial du/dx-partial idv/dy

[NB: u and v are the real and imaginary parts of the expanded function]

Question 24

what is another interpretation for differentiation

Answer 24

conformal mapping must be preserved, when thinking about differentiation as an evaluation of the function at a small change delta(Z) away. Must be mapped to a point that is stretched and rotated to be differentiable

delta(g)=delta(Z)g’(Z)
where Z is the argument of the function. This change gives a rotation and scale factor, which must be preserved.