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Example Questions
Example Question #21 : Complex Analysis
Given a complex number
, under what conditions is the following equation true?
The equation is always true.
The equation is only true if
.The equation is never true.
The equation is only true if
.The equation is only true if
.The equation is only true if
. denotes the conjugate of
and is defined as
.
Substituting this into the equation and simplifying yields:
So the equation is only true if
.Example Question #21 : Complex Analysis
What is the value of
, where is in radians?Not enough information is given.
The magnitude of a complex number
.
If
, then , so =1.Example Question #23 : Complex Analysis
Which of the following is equivalent to this expression?
None of these
None of these
Note that
lies in the first quadrant of the complex plane.
Any nonzero complex number
and
.
(We stipulate that is in radians.)
Conversely, a nonzero complex number
and
.
We can convert
,
and
Since
lies in the first quadrant of the complex plane, as does , .So
.
We now substitute this into our original expression and expand.
.
Finally, we convert this number back to the form
So our final answer is
.Example Question #21 : Complex Analysis
Which of the following is equivalent to this expression?
None of these
Note that
lies in the first quadrant of the complex plane.
Any nonzero complex number
and
.
(We stipulate that is in radians.)
Conversely, a nonzero complex number
and
.
We can convert
,
and
Since
lies in the first quadrant of the complex plane, as does , .So
.
We now substitute this into our original expression and expand.
.
Because , we substitute with the coterminal angle .
.
Finally, we convert this number back to the form
So our final answer is
.Example Question #24 : Complex Analysis
If
then what is the value of
?
None of these
Note that the
on the right side of the equation can be written as .
Multiplying the first two terms on the left side yields
.
Note that this number lies in the third quadrant of the complex plane.
We now convert
and
.
.
Since lies in the third quadrant of the complex plane, as does , .
So our new form is .
Our equation now reduces to
.
We solve for z by dividing.
Finally, we convert this to the form
and
.
.
So our final answer is
.Example Question #25 : Complex Analysis
Given a complex number
, under what conditions is the following equation true?
The equation is only true if
.The equation is never true.
The equation is only true if
.The equation is only true if
.The equation is only true if
.The equation is only true if
.
.
Substituting this into the equation and simplifying yields:
So the equation is only true if
.Certified Tutor
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