5. (9 points) Show that the function u(x, y) = x³ − 3xy² + y is harmonic and determine it's harmonic conjugate.

- Calculate \lim _{n \rightarrow \infty}\left(1+\frac{1}{\xi^{n}}\right)^{2^{n}}

) Find the Fourier series for the function f(x)=\left\{\begin{array}{cc} -1: & -\pi \leq x \leq 0 \\ 1: & 0<x \leq \pi \end{array}\right. defined on the interval [-, T]. (b) State the Intermediate Value Theorem and use it to show that f(x)=2 x^{4}-7 x^{2}+x+2 has four real solutions. (c) Let f(x)=\left\{\begin{aligned} x &: x \leq 0 \\ -x &: x>0 \end{aligned}\right. Show that f(x) is not differentiable at x = 0. Explain your answer carefully.

(a) Classify the following subsets of R as bounded or unbounded. If the set is bounded, write down the supremum and infimum. There is no need to provide proofs for your answers. \text { (i) }\{x:|x-7| \leq 3\} \text {. } \left\{\frac{3}{n^{2}}: n \in \mathbb{N}\right\} \bigcup_{n \text { prime }}\left(-\frac{1}{n^{2}}, \frac{1}{n^{2}}\right) Prove directly from the definition that the sequence \left\{\frac{3 n^{2}}{2 n^{2}+7}\right\} converges toNI CO32 Find the sum of the series \sum_{n=1}^{\infty} \frac{1}{n^{2}+8 n+15}

a) Evaluate the following contour integrals which are taken anticlockwise around the boundary of the disk |z| = 2 of radius 2 centred at the origin.Ensure you justify your answers. \text { (i) } \int_{|z|=2} \frac{e^{3 z}}{z-i} d z ) If C₁ is the straight line path from 0 to 1, C₂ is the straight line path from 1 to1 + i and C is the straight line path from 0 to 1 + i. i) Show C₁, C₂ and C on a labelled Argand diagram. \text { ii) Using the fact that } \int_{C} f(z) d z=\int_{t=a}^{b} f(z(t)) \frac{d z}{d t} d t \text { find: } \text { (B) } \int_{C_{2}} R e(z) d z Hence find: \int_{C_{1}+C_{2}} R e(z) d z where C₁ + C₂ indicates the total path C₁ followed by C₂. \text { ii) Find } \int_{C} R e(z) d z \text {. What do you conclude about } \int_{C} R e(z) d z \text { ? } \int_{|z|=2} \frac{e^{z}}{6 \pi i-2 z} d z \int_{\mid z=2} \frac{1}{z^{4}} d z \int_{C_{1}} R e(z) d z

a) Let C be a set, and A and B be disjoint sets. Find a bijection vh: CA × CB → CAUB¸ and show that it is well-defined, injective,and surjective. (b) Denote by |X| the cardinality of a set X. (i) Let A, B, C, and D be sets such that |A| = |B| and |C| =|D]. Prove that |Ac| = |Bd (ii) Find a pair of sets A and B so that |A| < |B|, but |AN| =|B]. Find an example where B is countably infinite. (Noproof is necessary.) (iii) Find two different explicit bijections f, g: 5N → 2N. By explicit, I mean recipes for converting elements of 5N into elements of 2N. Explain your answer graphically in terms of the trees for 2N and 5N.

5. Show that the series Σn=0 (a+n)(a+n+1) 1 where a > 0 is convergent.

Find the principal value of (l-i)4i (page - 103 ex-2(c) of the book)

2. (8 points) Evaluate the contour integral 24 dz +4 where C is the counterclockwise oriented curve | z |= 1.

2. (8 points) Show that the bilinear map w = maps points on the circle | z — 1 |= 1 to points on the line Re(w) = 12.