Complex Variable Functions: Test 1 Solutions Explained
School
University of Ottawa**We aren't endorsed by this school
Course
BIOL 2107
Subject
Mathematics
Date
Dec 11, 2024
Pages
2
Uploaded by BailiffAntMaster48
MATH 3007A - Functions of a Complex VariableTest 1 SolutionsOctober 6, 2016[Marks]1. Letz= 2-3i. Determine the real partRe(z), the imaginary partIm(z), the complex[4]conjugatez, and the magnitude|z|, ofz.Solution:Re(z) = 2,Im(z) =-3,z= 2 + 3i,|z|=√13.2. Express3-2i2 +iin the forma+bi,a, b∈R.[2]Solution:3-2i2 +i=3-2i2 +i2-i2-i=4-7i5=45-75i.3. Letz=-1-√3i.[3](a) Expresszin polar form.Solution:|z|= 2⇒z= 2parenleftBigg-12-√32iparenrightBigg= 2e-2πi/3= 2e4πi/3.(b) Determine arg(z) if arg(z)∈[0,2π).Solution:arg(z) =4π3.(c) Determine arg(z) if arg(z)∈[-π, π).Solution:arg(z) =-2π3.4. Letz= 3e-πi/3.[2](a) Determine log(z) if arg(z)∈[0,2π).Solution:log(z) = ln(3) +5πi3.(b) Determine log(z) if arg(z)∈[-π, π).Solution:log(z) = ln(3)-πi3.5. Letz=-2e2πi/3.[4](a) Determine log(z) if arg(z)∈[0,2π).Solution:log(z) = ln(2) +5πi3.
2(b) Determine log(z) if arg(z)∈[-π, π).Solution:log(z) = ln(2)-πi3.6. Find the square roots of-iand express them in the forma+bi,a, b∈R.[4]Solution:-i=e3πi2+2πik⇒the square roots areeπi4(3+4k),k= 0,1, i.e.,e3πi/4ande7πi/4.e3πi/4=-1√2+1√2iande7πi/4=1√2-1√2i.7. Find the cube roots of 64 and express them in the forma+bi,a, b∈R.[6]Solution:64 = 64e2πik⇒the cube roots are 4e2πik/3,k= 0,1,2, i.e., 4, 4e2πi/3=-2 + 2√3i,and 4e4πi/3=-2-2√3i.8. Find the fourth roots of 8√3 + 8i. You may leave them in polar form.[5]Solution:8√3 + 8i= 16eπi6+2πik⇒the fourth roots are 2eπi24(1+12k),k= 0,1,2,3, i.e.,2eπi/24, 2e13πi/24, 2e25πi/24, and 2e37πi/24.