Complex numbers

Question 1

What are the polar and exponential forms of x+iy?

Answer 1

r(cosØ+isinØ) and re^iØ

Question 2

“The derivative of sine equals cosine” is only valid for radians because of…

Answer 2

the step requiring substitution of 1 for lim(x->0) sinx/x

Question 3

For z=r(cosØ+isinØ)…

1. What is r called?
2. What is it equal to?

Answer 3

1. modulus or absolute value
2. sq(x²+y²) or sq(z_z)_

Question 4

For z=e^iØ, Ø is called the…

Answer 4

angle, phase, argument, or amplitude of z.

Question 5

How do you convert a fraction of complex numbers in rectangular form into a plain complex number in rectangular form?

Answer 5

Multiply the numerator and denominator by the complex conjugate of the denominator.

Question 6

What is the complex conjugate of a number z in all forms?

1. (x+iy)*=?
2. [r(cosØ+isinØ)]*=?
3. [re^iØ]*=?

Answer 6

1. x-iy
2. r(cosØ-isinØ)
3. re^-iØ

Question 7

The sum, difference, product, or quotient of conjugates is equal to…

Answer 7

the conjugate of the sum, difference, product, or quotient.

Question 8

To find the conjugate of an expression…

Answer 8

switch the signs of all the i terms.

Question 9

Given complex f and g,

1. |f·g|=?
2. |f/g|=?

Answer 9

1. |f|·|g|
2. |f|/|g|

Question 10

What is sinØ in terms of e?

Answer 10

(e^iØ-e^-iØ)/2i

Question 11

What is cosØ in terms of e?

Answer 11

(e^iØ+e^-iØ​)/2

Question 12

How are the exponential forms of sinØ and cosØ derived?

Answer 12

By writing Euler’s formula for e^iØ and e^-iØ and then solving for sinØ and cosØ.

Question 13

What is sinhØ in terms of e?

Answer 13

(e^Ø-e^-Ø)/2

Question 14

What is coshØ in terms of e?

Answer 14

(e^Ø+e^-Ø)/2

Question 15

sin(iz)=?

Answer 15

isinh(z)

Question 16

ln(re^iØ)=?

Answer 16

lnr+iØ

Question 17

cos(iz)=?

Answer 17

cosh(z)

Question 18

a^b=?

Answer 18

e^blna

Useful for finding powers of complex numbers

Question 19

a^b·a^c might have more answers than just a^b+c because…

Answer 19

a^b·a^c=e^blna·e^clna and lna can have more than one value for complex numbers.

Question 20

Answer 20

x²+y²