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Complex Analysis
Express sin 5θ and cos 5θ/ cos θ in terms of sin θ
Complex Analysis
Express −1 + i in polar form. Hence show that (−1 + i)16 is real and that 1/(−1 + i)6
is purely imaginary, giving the value for each.
is purely imaginary, giving the value for each.
Complex Analysis
Use De Moivre’s theorem to simplify the following
(a) (cos π/5+i sin π/5)10, (b) (cos π/9+i sin π/9)−3, (c) {cos(−π/6) + i sin(−π/6)}−4
(a) (cos π/5+i sin π/5)10, (b) (cos π/9+i sin π/9)−3, (c) {cos(−π/6) + i sin(−π/6)}−4
Complex Analysis
Express the roots of the equation z3 − α3 = 0 in terms of α and w, where w is
a complex cube root of unity. Use your answer to find the roots of the following
equations in the form a + ib.
a complex cube root of unity. Use your answer to find the roots of the following
equations in the form a + ib.
Complex Analysis
9.The principal value of the argument of (1+ ˆš3)(1−i) is____
\\(-5\\fra{\\pi}{12}\\)
\\(\\fra{5\\pi}{12}\\)
\\(\\fra{-\\pi}{12}\\)
\\(\\fra{\\pi}{12}\\)
\\(-5\\fra{\\pi}{12}\\)
\\(\\fra{5\\pi}{12}\\)
\\(\\fra{-\\pi}{12}\\)
\\(\\fra{\\pi}{12}\\)
Complex Analysis
Show that w=z+e^z is analytic and hence find dw/dz
Complex Analysis
9.3 Let w be a negative real number, z a 6
th root of w.
(a) Show that z (k) = ρ^
1
6
-
cos (pi+2kpi/6 )+ isin ( pi+2kpi/6), , k = 0, 1, 2, 3, 4, 5 is a formula for the
6th roots of w.
th root of w.
(a) Show that z (k) = ρ^
1
6
-
cos (pi+2kpi/6 )+ isin ( pi+2kpi/6), , k = 0, 1, 2, 3, 4, 5 is a formula for the
6th roots of w.
Complex Analysis
Given z = cos θ + isin θ and u + iv = (1 + z)(1 + z
2
). Prove that v = u tan( 3θ
2
) and
u
2 + v
2 = 16 cos2
(
θ
2
) cos2
(θ)
2
). Prove that v = u tan( 3θ
2
) and
u
2 + v
2 = 16 cos2
(
θ
2
) cos2
(θ)
Complex Analysis
Solve (-1)^2i
Complex Analysis
express cos^(3)tetasin^(4)teta in terms of multiple angle
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