Theorems

Question 1

Max Mod

Answer 1

Let D be a domain and f a holomorphic function from D to C. If there exists an a in D such that |f(a)| >=|f(z)| for all z in D then f is constant.

Question 2

Cauchy’s Residue theorem for simple closed contours

Answer 2

Let Y be a positively oriented simple closed contour and let f be meromorphic on DintY U Y. Assume f has no poles on Y and a finite number of poles in the interior of Y. Then the contour integral is 2pi * sum of the residues of those poles.

Question 3

Liouville’s Theorem

Answer 3

Every bounded entire function is constant.

Question 4

Fundamental theorem of algebra

Answer 4

Every non constant polynomial with complex coefficients has a complex root.

Question 5

Heine Borel for R or C

Answer 5

A subset K is compact if and only if K is closed and bounded

Question 6

Converse to FTC

Answer 6

Let f be a continuous function from a domain to C. If the contour integral of f is zero for all closed contours in D then f has an anti derivative.

Question 7

Estimation Lemme

Answer 7

The modulus of a contour integral is less than or equal to the length of the contour * supremum |f|

Question 8

Cauchy’s theorem for starlike domains

Answer 8

If D is a starlike domain and f holomorphic on D then the integral of f across any closed contour is 0.

Question 9

Union of open sets

Answer 9

The (possibly infinite) union of open sets is open.

Question 10

Intersection of open sets

Answer 10

Any intersection of finite open sets is open.

Question 11

Cauchy Riemann Equations

Answer 11

Let f be a complex diff function at z0. Then Ux = Vy , Uy=-Vx

Question 12

Principle of isolated zeros

Answer 12

For a function holomorphic on a ball size r around a and not identically zero there exists a rho between 0 and r such that f(z)=/= 0 for every z in the ball-{a}

Question 13

Cauchy’s Integral Formula

Answer 13

See notes

Question 14

Uniqueness of Analytic continuation

Answer 14

Let D’ contained in D be non empty domains and f be holomorphic on D’. Then there exists at most one holomorphic g on D such that f(z)=g(z) for every z in D’

Question 15

Weierstrass M test

Answer 15

fn is a sequence of functions such that the modulus of fn is less than Mn for all x in X and the infinite sum of Mn is finite. Then the infinite sum of fn converges uniformly.

Question 16

Test for determining removable singularity.

Answer 16

Let f holo on the punctured ball around a then f has a removable singularity at a if and only if lim(z-a)f(z)=0

Question 17

Jordan curve theorem

Answer 17

Let Y be a simple closed curve. Then its complement is a disjoint union of two domains, one of which is bounded,

Question 18

Identity Theorem

Answer 18

Let f and g be homomorphic on a domain D. If the set S containing the points where f=g contains a non isolated point then f = g on D.

Question 19

D2D

Answer 19

Every mobius trans from D to D is of the form MT with T in SU(1,1)

Question 20

Big Picard theorem

Answer 20

Let f be holomorphic on the punctured ball size R around a with an essential singularity at a. There is some point b in the complex plane such that for every 0

Question 21

Casorati Weierstrass

Answer 21

Let f be holomorphic on the punctured ball size r around a. with an essential singularity at a. Then for every w in C and for every r,epsilon>0 there exists a z in the punctured ball such that f(z) is contained in the punctured ball.

Question 22

Open Mapping theorem

Answer 22

Let D be a domain and f holomorphic on D and non constant. Then f maps open sets in D to open sets in C. More precisely if U in D is open then f(U) is open.

Question 23

Test for Uniform convergence

Answer 23

Uniform convergent if |fn(x) - f(x)|

Question 24

Test for non-uniform convergence

Answer 24

If there exists a sequence in X such that |fn(xn)-f(x)|>c for some positive constant c then fn doesnt converge uniformly.

Question 25

Rouché's Theorem

Answer 25

Let Y be a simple closed contour and f,g holomorphic on the Dint U Y. Suppose the modulus of the difference of f and g is less than the modulus of g for every z in Y then f and g has the same number of zeros.

Question 26

Riemann Extension theorem

Answer 26

Let f be holo and bounded on the punctured ball around a then f has a removable singularity at z =a

Question 27

Three point theorem

Answer 27

If we have two sets of 3 ordered distinct points then there exists a unique mobius trans mapping one set to the other.

Question 28

Morera's Theorem

Answer 28

f continuous on an open set U. if contour integral is 0 for all closed contours

Question 29

Morera's Theorem

Answer 29

f continuous on an open set U. if contour integral is 0 for all closed contours in U then f is holomorphic.