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Find the Taylor's theorem expansion of logz=(z-1) -(z-1) Β²/2+(z-1) Β³/2-........., when |z-1|<1.
Find the value of β«c 1/πππ§, where C is circle π§=π^πβ , 0β€β β€π
Expand f(z)=π§+3/π§(π§2βπ§β2) in power of z where
a) |π§|<1
b) 1<|π§|<2
c) |π§|>2
And z (z = x +yi : x,y are real numbers) is a complex numbers satisfies the condition 2|(x+yi)-1-2i| = |3i + 1 - 2(x-yi)|
Expand f(z)=
"Expand f(z)= \ud835\udc67+3\ud835\udc67(\ud835\udc672\u2212\ud835\udc67\u22122)\nin power of z where\na) |\ud835\udc67| < 1\nb) 1< |\ud835\udc67| < 2\nc) |\ud835\udc67| > 2"π§+3 π§(π§ 2βπ§β2) in power of z where a) |π§| < 1 b) 1< |π§| < 2 c) |π§| > 2Β
Let F, G be meromorphic functions such that Fn+Gn=1, assuming nβ₯4 if necessary. Prove that F and G are constants.
(8.1) Let z = z1/z2 where z1 = tan ΞΈ + i and z2 = z1. Find an expression for z n with n β N.
(8.2) Let z = cos ΞΈ β i(1 + sin ΞΈ). Determine 2z + i / β1 β iz
Use De Moivreβs Theorem to
(7.1) derive the 4th roots of w = β8i
(7.2) express cos(4ΞΈ) and sin(5ΞΈ) in terms of powers of cos ΞΈ and sin ΞΈ
(7.3) expand cos6ΞΈ in terms of multiple powers of z based on ΞΈ
(7.4) express cos3ΞΈ sin4ΞΈ in terms of multiple angles.
Determine for which value (s) of Ξ» the real part of z = 1+Ξ»i/1βΞ»i equals zero
Find the roots of the equation:
(5.1) z4 + 4 = 0 and z4 β 4 = 0
(5.2) Additional Exercises for practice are given below.
Find the roots of
(a) z8 β 16 = 0
(b) z8 + 16 = 0.Β
