Complex Analysis Answers

obtain first three terms of taylors series for f(z) = sin z about 2=0

Obtain first three terms of taylors series for f(z) = sin z about z =0

Find the square root of −5 − 12𝑖

Evaluate ∫f 𝑜𝑣𝑒𝑟 𝑐 where 𝑓( 𝑧 )= 𝑥^2 + 𝑖𝑦^2 where c is given by 𝑧 (𝑡 )= 𝑡^2 + 𝑖𝑡^2, 0 ≤ 𝑡 ≤ 1

Show that 𝑠𝑖𝑛𝑖𝑦 = 𝑖 𝑠𝑖𝑛hy

Using the Cauchy –Riemann equations verify the following is analytic or not

i) 𝑥^2 − 𝑦^2 + 2𝑖𝑥𝑦

ii) 𝑥^2 + 𝑦^2 − 2𝑖𝑥𝑦

find the residue of f(z)=z^2+2z/(z+1)^2(z+4)at its poles

z₁=3.45∠980°

z₂₌-5+9i

1. z1+z2 final answer in polar form
2. z1-z2 final answer in trigonometric form
3. z2*z1 final answer in exponential form
4. z1/z2 final answer in rectangular form

1a) Find the real numbers a and b such that

i5(3 - 2i)2 - (a - bi) = a - bi.

b. Let z1 =  √3 + i and z2 = √3 - i. Show that

z₁⁶¹ + z₂⁶¹ = 2⁶¹√3.

(Remember that: cos(- θ) = cos(θ), sin(θ) = -sin(θ) = -sin(θ), cos(2kπ + θ) = cos(θ) and sin(2kπ + θ) = sin(θ) )