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Exercises: Complex Numbers, Vectors, and Functions

Module by: Steven Cox

Summary: These are exercises to practice topics covered in modules regarding complex numbers, vectors, matrices, complex functions, and complex differentiation.

Exercises

Exercise 1

Express |ⅇz| z in terms of xx and/or yy.

Solution 1

Pending completion of assignment.

Exercise 2

Confirm that ⅇlnz=z z z and lnⅇz=z z z .

Solution 2

Pending completion of assignment.

Exercise 3

Find the real and imaginary parts of coszz and sinzz. Express your answers in terms of regular and hyperbolic trigonometric functions.

Solution 3

Pending completion of assignment.

Exercise 4

Show that cos2z+sin2z=1 z 2 z 2 1

Solution 4

Pending completion of assignment.

Exercise 5

With zw≡ⅇwlnz z w w z for complex zz and ww compute ⅈ

Solution 5

Pending completion of assignment.

Exercise 6

Verify that coszz and sinzz satisfy the Cauchy-Riemann equations and use the proposition to evaluate their derivatives.

Solution 6

Pending completion of assignment.

Exercise 7

Submit a Matlab diary documenting your use of residue in the partial fraction expansion of the transfer function of B=200-1400-12 B 2 0 0 -1 4 0 0 -1 2 .

Solution 7

Pending completion of assignment.

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