deck 1

Question 1

f is continuous at z if and only if ?

Answer 1

limf(z) = f(z_o)

Question 2

f is differentiable at x if and only if ?

Answer 2

-

Question 3

state the Cauchy equations

Answer 3

-

Question 4

Cauchy thm. f being differentiable implies two things

Answer 4

partial derivertives exist and cauchy eqns hold

Question 5

The converse thm ?

Answer 5

partial derivertives exist and continuous and cauchy eqns hold THEN f is differentiable

Question 6

what is the ratio test of a power series?

Answer 6

-

Question 7

what is the root test of a power series?

Answer 7

-

Question 8

what is the sum formula of exp(x)

Answer 8

Σ (z^n/n!)

Question 9

what is the sum forumla for cos(z)?

Answer 9

-

Question 10

what is the sum formula for sin(z)?

Answer 10

-

Question 11

log(z) = ?

Answer 11

ln|z| + iarg(z)

Question 12

Log(z) =?

Answer 12

ln|z| + iArg(z)

Question 13

cos(A+B) = ?

Answer 13

cosAcosB - sinAsinB

Question 14

sin(A+B) = ?

Answer 14

cosAsinB + cosBsinA

Question 15

what is the contour integration formula?

Answer 15

-

Question 16

contour is closed if ?

Answer 16

γ(a)=γ(b)

Question 17

lengh(γ) = ?

Answer 17

-

Question 18

what is the fundamental theorem of contour integration?

Answer 18

-

Question 19

what is the winding number? which rotation is negative ?

Answer 19

-

Question 20

ω(r,0) = ?

Answer 20

-

Question 21

ω(r,z_o) = ?

Answer 21

-

Question 22

what is the estimation lemma?

Answer 22

-

Question 23

what is Cauchey’s theorem? (not generalised)

Answer 23

-

Question 24

what is Cauchey’s generalised theorem ?

Answer 24

-

Question 25

what is Cauchey's integral formula for a circle?

Answer 25

-

Question 26

what is Taylor's theorem

Answer 26

-

Question 27

what is caucheys estimate

Answer 27

-

Question 28

what is liouvilles theorem ?

Answer 28

-

Question 29

what is the fundamental theorem of algebra

Answer 29

-

Question 30

what is laurents theorem ?

Answer 30

-

Question 31

what is a singularity?

Answer 31

-

Question 32

what is a pole and its order?

Answer 32

-

Question 33

if f(z) = p(z)/q(z) when i and ii hold, then f has a pole of order m at z_o. what are the conditions i, ii ?

Answer 33

-

Question 34

Res(f,z_o) = ?

Answer 34

b_1

Question 35

Res(f,z_o) also equals ? hint answer is an integral

Answer 35

-

Question 36

what is caucheys residue theorem?

Answer 36

-

Question 37

if f(z) = p(z)/q(z), and more conditions, what are they? then Res(f,z_o) = ?

Answer 37

-

Question 38

dz = ?

Answer 38

iexp(it)

Question 39

if the integrals of z_o to z_1 are different on different paths, what does this imply?

Answer 39

-

Question 40

what is the sum formula of 1/(1-z)

Answer 40

-

Question 41

outline the comparison test

Answer 41

-

Question 42

b^a = ?

Answer 42

exp(aln(b))

Question 43

a smooth path is ?

Answer 43

differentiable

Question 44

what does Res(f,z_o) also equal ? hint answer is a limit

Answer 44

-

Question 45

define a singularity

Answer 45

-

Question 46

define the principle part of the laurent series

Answer 46

-

Question 47

what is the integral from -∞ to ∞ f(x)dx defined to be ? hint its a limit. and what is the principle value ?

Answer 47

-

Question 48

what is the formula that ensures the principle value is equal to the defined value?

Answer 48

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