1 + z 1 x (2x y)e x2 xy. xe x2 xy. x x3 e x, lim x log(x). (3 + i) 2 17i + 1. = 1 2e + e 2 = cosh(1) 1 + i, 2 + 3i, 13 exp i arctan

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1 Complex Analysis I MT333P Problems/Homework Recommended Reading: Bak Newman: Complex Analysis Springer Conway: Functions of One Complex Variable Springer Ahlfors: Complex Analysis McGraw-Hill Jaenich: Funktionentheorie Springer Arnold: Complex Analysis Compute the Jacobi matrix of the map f : R 3 R 2 with fx y ) = x + y + x e x2 xy ). d xy) f = + x 2x y)e x2 xy xe x2 xy ) 2. Compute 2 3. Compute the limits Both limits are. please hand up -3 Monday 29/9 in class 4. Express the following in the form a + ib with a b R: d t dt x 34 + x dx. t= t lim x x3 e x lim x logx). x 3 + 7i)3 + 3i) 3 + 7i)3 + 3i) = i9 + 2) = 2 + 3i 3 + i) 2 7i + cosi). 3 + i) 2 7i + = 3 + i)2 7i) 7i + ) 7i) = 3 + i)2 7i) 3i = Compute modulus and an argument for cosi) = eii + e ii 2 = 2e + e 2 = cosh) + i 2 + 3i 3 2 i 2. + i = 2 exp i π ) i = )) 3 3 exp i arctan i 2 = exp i π ) 3 6. Write the polynomial px) = x R[x] as a product of quadratic/linear real polynomials. Abbreviating j := exp iπ 4 ) hence j2 = i we compute x = x 2 2i)x 2 + 2i) = x + 2j)x 2j)x + 2ij)x 2ij) = x + 2j)x 2ij) x + 2ij)x 2j) = x 2 + 2x + 2) x 2 2x + 2) please hand up 4-6 Monday 6/ in class 7. For the following functions u: U C R find functions v : U R such that f = u + iv is holomorphic on U: ux + iy) = x + x 2 y 2 U = C and vx y) = y + 2xy vx y) = arctany/x) ux + iy) = lnx 2 + y 2 ) U = {x + iy C x y R x > }

2 8. Determine all harmonic two variable polynomials px y) R[x y] of degree 3. Hint: A polynomial px y) R[x y] of degree 3 has the form px y) = p ij x i y j in total coefficients p ij. ij i+j 3 9. Describe the sets { e x+iy x y R < x < < y < π } and { e x+iy x y R < x < π 4 < y < 3π } 4 by inequalities. { C < < e I) > } { C < < e I) > R) } please hand up 7-9 Monday 3/ in class. Let f : C C be the function with f) = for all C. Compute f) d = d for the following curves = i : a) : [ 2π] t) = e it t : t b) 2 : [ 3] 2 t) = + it ) : t 2 3 t + i3 t) : 2 t 3 c) 3 : [ ] 3 t) = + tv for some v C. More generally prove that d is imaginary whenever is closed. d) 4 : [ ] 4 t) = t + it 9 t). Hint: If you can say guess from a b c) what d means geometrically for a closed curve you need not rigorously prove your answer to d. 2πi i v 8i/5. The integral is 2i times the area of the interior of the curve.. Let U C and µ: [ 2] U be smooth curves given by U = { C R) I) } and t) = + t µt) = i + it. Find a smooth map H : [ ] [ ] U such that H t) = t) and H t) = µt). Hs t) = + t)e iπs Describe the following regions in C by inequalities for I) R) : a) U = {x + iy) 2 x y R + } b) U 2 = {sinx + iy) x y R} c) U 3 = { x+iy x y R x y > }. 3. The lenght of a C -curve : [a b] R n is defined as length) = b a t) dt. a) For λ R r b > compute the length of the helix : [ b] R 3 t) = r cost) r sint) λt). b) Prove that if [a b] U C is C then f) d length) max{ f) U} for any continuous function f : U C. c) Assume that f is a holomorphic function on B ) with f) = and f ). Prove that for sufficiently small r > we have f) d = 2πi f ). =r Hint: Prove first that /f is holomorphic on B r )\{} for sufficiently small r and then that the integral does not depend on r provided is is small. You might use that f) = f q) ) + q) with some q such that lim = exists. 2π ire it 2π ire it d d =r f) r =r f) r fre it dt ) r f )re it + qre it ) dt 2π r i f ) + qreit ) re it dt = 2π i f ) + lim r qre it ) re it dt = 2πi f ).

3 4. Let V be a vector space over R with scalar product denoted by. The norm of a vector v V is the real number v = v v and the angle between two vectors v w V \ {} is defined to be the number v w) [ π] such that cos v w) = v w v w. A linear map A HomV V ) is called conformal if it preserves angles i.e. if Av Aw) = v w) for all v w V \ {}. a b a) Let V = R 2 with the standard scalar product. Determine all conformal 2 2-matrices c d b) Let A HomV V ) be Hermitian i.e. ) HomR 2 R 2 ). Av w = v Aw for all v w V and assume that A is conformal. How many different eigenvalues can A have? 5. Let U C be open and f : U C be holomorphic. Prove that the map is holomorphic. g : U := { U} C f) f) f) f) f ) f) f ) fu) f ) lim = f u u ) 6. Assume r > and that the complex numbers a k b k k N are such that the series k= a k k k= b k k converge for all B r ) to the same number i.e for all B r ). a) Prove that a k = b k for all k N. a k = f k) )/k! = g k) )/k! = b k. f) := a k k = b k k =: g) k= b) Let f : U = C \ {i i} C be the function with f) = e + 2 for all U. What is the radius of convergence of the Taylor series f k) 7 + 2i) k k! k= k= of f at 7 2i? The maximal r such that f is holomorphic on B r 7 2i) is r = min{ 7 2i i 7 2i+i } = 7 i = 5. please hand up -6 Wednesday 2/ in class 7. Let f : B / /4) C be the function given by f) = ) 3 + ) What is the radius of convergence of the Taylor series of f at /4? 8. Compute the Taylor series at of the function f given by the formula f) = Hint: You need to give a formula for a k C such that f) = k= a k k. for all B / /4). 9. Compute the first four terms of the Taylor series at of the function f with f) = sine ). please hand up 7-9 Wednesday 26/ in class. 2. Find the Laurent series of the function around which converges near 2. f : C \ { } C with f) = for all C \ { }

4 2. Classify the isolated) singularities of the following functions f i as removable pole essential. Also give the order of the poles. a) f : C \ { } C f) = ) 3 b) f : C \ {} C f) = cos) sin2 ) 5 ) c) f : C \ {i i} C f) = sin Let f : C C be holomorphic. Assume that Compute f i). please hand up 2-22 Monday 8/2 in class 23. Prove that if sin) = for some C then /π Z. Hint: sin) = e i e i) /2i f) 7 for all C and f 7) ) = 7!i. 24. Classify the isolated) singularities of the following functions f i as removable pole essential. Also give the order of the poles. a) f : C \ Z C f) = 2 sin) 3 b) f : C \ { } C f) = e2 2 2 ) c) f : C \ { kπ k Z \ {}} C f) = sin/). 25. Determine all meromorphic functions f : C C with f) < e / for all C where f is defined. Hint: What singularities can f have? You might consider looking at the Laurent series. 26. a) Prove that a non-constant meromorphic function f : C C has at most finitely many poles and eros in B ). b) Let f : C C be a meromorphic function and assume that the function f : C C with f) = f/) for all C where f/) is defined is also meromorphic. Prove that f is a rational function i.e. that there are p) q) C[] such that f) = p)/q) for all C which are not a pole of f. Hint: The function / maps B ) to C { } \ B ). In C a bounded sequence has a convergent subsequence. please hand up Monday 5/2 in class 27. Write the following complex numbers in the form = a + ib and = re iφ with suitable real numbers a b r > φ. = + i i = 2 + 3i) 4 and = e +i. 28. Find the Laurent series expansion around of the function f : C \ B 2 ) C f) = ) ) for all C \ B 2 ) which converges in a neighbourhood of Find a function v : C R such that f = u + iv : C C is holomorphic where u: C R is the function with u) = R)) 4 for all C. 3. The following formulas give a meromorphic function f : C C. Find all poles of f and determine their order. f) = sin) 2 4 f) = sin) cos)) 3 f) = sin) 4 π Decide which of the isolated singularities of the following functions f : C \ S C are removable singularities poles essential singularities. ) S = { 2 2} f) = sin 2 4) S = { } f) = e 2 2 ) + )

5 32. Assume that the series k Z a k3 i) k converges and that k Z a k i) k = sin) for all B /888 3). Find the smallest possible r R r > and the largest possible R R such that k Z a k i) k converges for all C with r < i < R. 33. Determine all meromorphic functions f : C C such that f) e 2 for all C with >.