MAT135-Assignment10

# MAT135-Assignment10 - i √ 3 for z ∈ C(d Solve sin z = i...

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MATH 135, Fall 2010 Assignment #10 Due at 4:00pm on Tuesday, November 30 Problem 1 . If z,w C , prove that | z + w | 2 + | z - w | 2 = 2 | z | 2 + 2 | w | 2 . Problem 2 . For real numbers x and y , we deﬁne e x + iy = e x cos y + ie x sin y . For a complex number z , we deﬁne cos z = e iz + e - iz 2 and sin z = e iz - e - iz 2 i . (a) Show that for all z,w C we have e z + w = e z e w . (b) Show that for all z,w C we have sin( z + w ) = sin z cos w + cos z sin w . (c) Solve e z = 1 +
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Unformatted text preview: i √ 3 for z ∈ C . (d) Solve sin z = i for z ∈ C . Problem 3 . Draw a picture of each of the following subsets of the plane. (a) ± z ∈ C ² ² 1 < | z-1 | ≤ √ 5 ³ (b) ± z ∈ C ² ² | z-2 i | = | z-4 | ³ (c) ± z ∈ C ² ² | z | + | z-4 | = 8 ³ 1...
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