We reviewed for the comprehensive final exam to be given 7:30–9:30 on the morning of Friday, December 9.
December 1
We discussed zeroes of analytic functions, the identity principle, and applications to trigonometric identities.
The assignment for next time (not to hand in) is to make a list of the major topics from the semester.
November 29
We worked out the improper real integral
\[\int_0^\infty \frac{1}{1+x^4}\,dx\]
by applying the residue theorem to an appropriate closed curve. Then we started working on the following two problems, which are a take-home quiz due next class.
Use the residue theorem with a semicircular path to show that \[ \int_0^\infty \frac{1}{1+x^2}\,dx =\frac{\pi}{2}\]
(a result that you already know how to get from real calculus).
Use the residue theorem to show that \[\int_0^\infty \frac{x}{1+x^3}\,dx = \frac{2\pi}{3\sqrt{3}}\]
(a result that is harder to get by real calculus methods). For this problem, an appropriate path is the edge of one-third of a pie.
November 22
In class, we learned a new method for computing residues, we applied the residue theorem to compute the integral
\[\int_0^{2\pi} \frac{1}{5+3\cos(\theta)}\,d\theta,
\]
and we worked the following three problems in groups for a quiz grade:
Find the residue at \(0\) of \(\dfrac{e^{2z}} {\sin(3z)}\).
Find the residue at \(2i\) of \(\dfrac{z}{z^2+4}\).
The answers are \(1/3\), \(1/2\), and \(2\pi i\).
There is no assignment over the Thanksgiving break.
November 17
Here is the assignment to hand in next time. (This assignment is partly a reminder that you should be starting to review for the final exam.)
Problem 15(d) on page 12, which asks you to calculate the quantity
\[\frac{(-1+\sqrt{3}\,i)^{15}}{(1-i)^{20}} + \frac{(-1-\sqrt{3}\,i)^{15}} {(1+i)^{20}}.
\]
Remark: You can save some work if you notice that this expression is the sum of a certain complex number and its complex conjugate.
Problem 20 on page 45, which asks you to find an entire linear transformation with fixed point \(1+2i\) carrying the point \(i\) into the point \(-i\).
Remarks: The terminology is defined in Problem 16 on the preceding page. You are looking for a function \(f(z)\) of the form \(az+b\), where \(a\) and \(b\) are certain complex numbers, such that \(f(1+2i)=1+2i\) and \(f(i)=-i\). Since you have two unknowns \(a\) and \(b\), and you have two equations, you should be able to solve for \(a\) and \(b\).
Problem 6 on page 104, which asks you to find the radius of convergence of the power series
\[z+\frac{z^4}{2!} + \frac{z^9}{3!}+\frac{z^{16}}{4!} +\cdots+\frac{z^{n^2}}{n!}+\cdots.
\]
Remarks: This problem is tricky not only because of the factorials but because the series is missing many powers of \(z\); the exponent on the general term is \(n^2\), not \(n\). One possible approach is to combine the comparison test with the root test, observing that the value \(n!\) is between \(1\) and \(n^n\).
In class, we looked at some more examples of computing residues and stated the residue theorem. Then we took the following quiz.
Find the residue at \(0\) of each of the following functions.
\(\dfrac{\cos(z)}{z^2}\)
\(\dfrac{z}{\sin(z)}\)
\(z^2 e^{1/z}\)
The first two problems both have answer equal to \(0\) (the easy method is to notice that both functions are even, so the term \(1/z\) cannot appear in the Laurent series), and the third problem has answer \(1/6\).
November 15
In class, we looked at some additional tricks, such as long division, for computing Laurent series, and we talked about the notion of the residue of an analytic function at a singular point.
Here is the assignment for next time (not to hand in, but be prepared for a quiz).
Problem 4(c) on page 167, which asks for the principal part of the Laurent expansion of the function
\[ \frac{z-1}{\sin^2(z)}\]
at the point \(0\).
Remarks: The standard notation \(\sin^2(z)\) is an abbreviation for \((\sin(z))^2\). One way to proceed is to write the Taylor series for \(\sin(z)\), square it, divide out a factor of \(z^2\), and do long division. You can verify your answer at WolframAlpha.
Problem 4(d) on page 167, which asks for the principal part of the Laurent expansion of the function
\[\frac{e^{iz}}{z^2+b^2}\]
at the point \(ib\), where \(b\) is some positive real number.
Suggestion: The denominator factors as \((z-ib)(z+ib)\), and you are looking for a Laurent series in positive and negative powers of \((z-ib)\). One way to get started (not the only way) is to split up the fraction \(1/(z^2+b^2)\) using partial fractions. You can verify your answer at
WolframAlpha.
Problem 21(h) on page 169, which asks for the residue at \(0\) of the function
\[z^n \sin(1/z)\]
when \(n\) is an integer.
Hint: The answer depends on \(n\). Consider cases: \(n\) might be even or odd, positive or negative.
November 10
Here is the assignment to hand in next time.
Problem 9 on page 150, which says to prove that
\[\frac{1}{1+z+z^2} = \sum_{n=0}^{\infty} \left( z^{3n} -z^{3n+1}\right)
\]
when \(|z|\lt 1\).
[The hint in the back of the book amounts to observing that \(1-z^3=(1-z)(1+z+z^2)\). Observe too that you can expand \[\frac{1}{1-z^3}\] in a geometric series when \(|z|\lt 1\).]
Problem 14(c) on page 151, which says to find the first four terms of the Taylor expansion at \(z=0\) of the function \(e^{z\sin(z)}\).
Suggestion: You know a series for the sine function, so you can easily get a series for the product \(z\sin(z)\), and you know a series for the exponential function. To get a series for the composite function, substitute one series into the other and expand. There is little hope of getting a general formula for the coefficients, but you are asked for just the first four terms, so you do not need a general formula. You can verify your answer at WolframAlpha.
Problem 4(a) on page 167, which asks for the principal part of the Laurent expansion of the function
\[\frac{z}{(z+2)^2}\] at the point \(-2\).
Amplification: The “principal part” of a Laurent series means the sum of the terms that have negative exponents. (So a Taylor series has no principal part.) For example, the Laurent series
\[ \frac{1}{(z+2)^2} +1+(z+2)^2+(z+2)^4+\cdots\]
with center \(-2\) has principal part \[\frac{1}{(z+2)^2}.\]
One way to solve your problem is to expand the numerator in a Taylor series in powers of \((z+2)\) and then divide out the denominator to get a Laurent series. You can verify your answer at
WolframAlpha.
In class, we looked at some more examples of Laurent series and applications to computing integrals. Then we did the following quiz.
For
the function \(\dfrac{3}{z(z-3)}\),
find the coefficient of \(\dfrac{1}{z}\) in the Laurent series
that converges in the punctured disk where \(0\lt |z|\lt 3\);
find the coefficient of \(\dfrac{1}{z}\) in the Laurent series
that converges in the region where \(3\lt|z|\);
find the coefficient of \(\dfrac{1}{z-2}\) in the Laurent series
that converges in the annulus where \(1\lt |z-2|\lt 2\).
[The answers are \(-1\), \(0\), and \(1\).]
November 8
We computed some Taylor series using geometric-series trickery, and we extended the method to obtain some examples of Laurent series (in positive and negative powers of \(z\)).
The assignment for next time (not to hand in) is Problem 2 on page 167, which says to find Laurent series expansions for the rational function
\[\frac{1}{(z-a)(z-b)}\]
(where \(a\) and \(b\) are unspecified values such that \(0\lt |a|\lt |b|\)) in the following four cases.
A series in powers of \(z\) valid in a disk centered at \(0\). (The book says to delete the center point of the disk, but this restriction is unnecessary. The function is analytic at \(0\), so the series will be an ordinary Taylor series.)
A series in powers of \(z-a\) and \(1/(z-a)\) valid in a punctured disk of the form \(0\lt |z-a|\lt r\). (The book calls such a region a “deleted disk,” meaning that the center point has been removed.) The radius \(r\) will be \(|b-a|\), which is the distance from the center point \(a\) to the singular point \(b\).
A series in powers of \(z\) and \(1/z\) valid in the annulus where \(|a|\lt |z|\lt |b|\).
A series in powers of \(1/z\) valid in the region where \(|z|\gt |b|\). (This region can be viewed as a degenerate annulus with outer radius infinity.)
Notice that answers to parts (b) and (d) are in the back of the book.
November 3
Here is the assignment to hand in next time.
Problem 2 on page 150 of the textbook: The power series
\(\displaystyle \sum_{n=1}^\infty \frac{z^n}{n}\) represents what function?
Problem 4 on page 150 of the textbook: Find the Taylor
expansion of the exponential function \(e^z\) centered at the point
where \(z=a\) [in other words, the series in powers of \((z-a)\)].
Problem 8 on page 150 of the textbook: Find the Taylor
expansion of the function \[\frac{1}{(z+1)(z-2)}\]
at the origin (that is, the series in powers of \(z\)).
[The hint in the back of the book is to start with the partial-fractions decomposition of this function.]
In class, we discussed the complex logarithm function and Taylor
series. We took the following quiz in groups:
Find all solutions \(z\) of the equation \(e^z=i\).
[Answer: \(z=(\pi i/2)+2n\pi i\), where \(n\) is an arbitrary integer.]
Find a complex number \(w\) such that \(i^w=-e\).
[One answer is \(2-(2/\pi)i\).]
Find the Taylor series of \(\log(z)\) centered at the
point \(1\), and determine the radius of convergence of this
series.
[Answer: The series is \(\displaystyle \sum_{n=1}^\infty
\frac{(-1)^{n+1}}{n} (z-1)^n\), since the constant term is equal to
\(0\). The radius of convergence is \(1\), since the series converges when \(|z-1|\lt 1\) and diverges when
\(|z-1|\gt 1\). In fact, this series is part of Problem 1 on
page 149 of the textbook. (The book writes “ln”
instead of “log” for the complex logarithm function.)]
November 1
In class, we derived the following formula for the complex logarithm function:
\[\log(z)=\log|z|+i\arg(z),
\]
where “log” on the right-hand side means the natural logarithm function (sometimes written “ln”).
Then we used Cauchy's integral formula for derivatives to establish Cauchy's inequalities for derivatives, which we applied to prove Liouville's theorem about entire functions.
Here is the assignment for next time (not to hand in, but expect a quiz).
Show that \(\log(z)\) is an analytic function by checking the Cauchy–Riemann equations.
Suggestion: Use the polar-coordinate form of the
Cauchy–Riemann equations from the assignment of September 13. What restriction needs to be imposed on the variable \(z\)?
For the complex logarithm function, is the property that
\[\log(z^4)=4\log(z)\]
always true?
Once you know the complex logarithm function, you can define complex powers of complex numbers by declaring \(z^w\) to mean \(e^{w\log(z)}\). Use this rule to evaluate the powers \(i^{(1+i)}\) and \((-i)^i\).
Are your answers compatible with what your calculator (or WolframAlpha) gives?
We reviewed for the exam to be given next class, and we discussed the previous homework assignment.
October 20
In class, we discussed the previous assignment and some related topics. Reminder: The second exam takes place on Thursday, October 27.
The assignment to hand in next time is to determine the image, under each of the following analytic
functions, of the square grid shown in the figure.
\(e^z\)
\(\cos(z)\)
\(\sin(z)\)
October 18
In class, we discussed the complex exponential, sine, and cosine functions and their power series.
Here is the assignment for next time (not to hand in).
Complete the solution of the problem we worked on at the end of class, to find all possible values of the complex number \(z\) when \(\sin(z)=4\).
My answer is \(z=\frac{\pi}{2}+2n\pi \pm i\log(4+\sqrt{15}\,)\), where \(n\) is an arbitrary integer, and \(\log\) denotes the real natural logarithm function. To simplify the answer, I used the observation that \(1/(4-\sqrt{15}\,) = 4+\sqrt{15}\).
Problem 5 on page 120, which says: “Describe the limiting behavior of \(e^z\) as \(z\to\infty\) along the ray \(\arg z=\alpha\).”
Problem 8(a) on page 120, which says to show that \(|\cos(z)|^2 = \cosh^2(y)-\sin^2(x)\).
(This statement is equivalent to the one in the book after squaring both sides.)
October 13
In class, we looked at examples of testing series for convergence, and we discussed the comparison test and two versions of the root test. We did not have time to take a quiz.
The assignment to hand in next time is to determine the radius of convergence of each of the following power series. Part of the problem is to decide which convergence test to apply.
Notice that in the second problem, no odd powers of \(z\) are present, and in the third problem, even more powers of \(z\) are missing (since the exponent on \(z\) is a power of \(2\)). One way to think about this situation is that many of the coefficients in the power series are equal to \(0\).
October 11
In class, we started a new topic: infinite series of complex numbers. We looked at examples involving alternating series and geometric series.
Here is the assignment for next time (not to hand in).
The ratio test for convergence of a series \(\displaystyle \sum_{n=0}^\infty c_n\) says
if \(\displaystyle \lim_{n\to\infty} \left| \frac{c_{n+1}}{c_n}\right| \lt 1\), then the series \(\displaystyle\sum_{n=0}^\infty c_n\) converges;
if \(\displaystyle \lim_{n\to\infty} \left|
\frac{c_{n+1}}{c_n}\right| \gt 1\), then the series \(\displaystyle\sum_{n=0}^\infty
c_n\) diverges;
if \(\displaystyle \lim_{n\to\infty} \left|
\frac{c_{n+1}}{c_n}\right| =1\), then try a different test;
if \(\displaystyle \lim_{n\to\infty} \left|
\frac{c_{n+1}}{c_n}\right| \) does not exist, then try a different test.
Since the computation in the ratio test involves taking absolute values, and since an absolutely convergent series converges, the ratio test applies both to real series and to complex series.
What does the ratio test tell you about the following series?
\(\displaystyle \sum_{n=0}^\infty \frac{3^n}{n!}\) (Note: by convention, \(0!\) equals \(1\).)
The radius of convergence of a power series \(\displaystyle\sum_{n=0}^\infty c_n z^n\) is the radius \(R\) with the property that the series converges when \(|z|\lt R\) and diverges when \(|z|\gt R\). For instance, the geometric series \(\displaystyle \sum_{n=0}^\infty z^n\) has radius of convergence equal to \(1\).
What does the ratio test tell you about the radius of convergence of the following power series?
Here is the assignment to hand in next class. These are problems that require more thought than computation, so your solutions should involve more words than formulas.
Problem 15 on page 74, which says to show that
\[\int_C \, \frac{1}{z^2+1}\,dz=0\]
for every simple closed curve \(C\) that is contained in the annulus where \(1\lt |z|\lt R\) (the number \(R\) being some unspecified radius greater than \(1\)).
Remarks: The book says the curve is “piecewise smooth”—meaning that the curve might have corners, like a rectangle—and is a “closed Jordan curve”—which is what we have been calling a simple closed curve. In essence, we have already done this problem in class (and in more than one way), so you basically need to interpret your notes for the grader.
Problem 18 on page 74, which says to describe the behavior of the integral
\[\int_{|z-a|=R}\, \frac{z^4+z^2+1}{z(z^2+1)}\,dz\]
as a function of the radius \(R\).
Remarks: You may assume that the point \(a\) lies on the real axis. Also assume that the circle is oriented in the standard counterclockwise direction. In view of the principle about invariance of the integral under deformation of the curve within a region where the integrand is analytic, the value of the integral should be a piecewise constant function of \(R\). The issue is to determine the values of \(R\) where the transitions occur and to determine the different values of the integral.
Problem 19 on page 75, which says to find all possible values of the integral
\[\int_C \,\frac{1}{z^2+1}\,dz\]
when \(C\) is a curve with initial point \(0\) and final point \(1\). What restriction must be imposed on \(C\)?
Remarks: If the path \(C\) goes along the real axis from \(0\) to \(1\), then you can compute the integral as a real calculus integral by using an antiderivative. On the other hand, if the path goes from \(0\) to \(-2\) along the real axis, then along a semicircle to the point \(2\), and then along the real axis back to \(1\), you will get a different answer, as you can see by adding and subtracting a path along the real axis from \(1\) to \(0\) to create a closed path to which you can apply Cauchy's integral formula. With these two cases in hand, can you use the path deformation principle to handle an arbitrary curve that joins \(0\) to \(1\)?
In class, we looked at further examples of computing integrals by applying Cauchy's integral formula, and we saw an example of evaluating a complex line integral by using an antiderivative. We also took the following quiz.
Let \(C\) be the circle defined by the equation \(|z|=2\), oriented in
the standard counterclockwise direction. Determine the value of each
of the following integrals.
\(\displaystyle \int_C\, \frac{z}{z-1}\,dz\)
\(\displaystyle \int_C \,\frac{z-1}{z^2}\,dz\)
\(\displaystyle \int_C \,\frac{z}{z^2-1}\,dz\)
(All three answers turn out to be \(2\pi i\).)
October 4
In class, we continued the discussion of Cauchy's integral
formula and deduced a version of the formula for derivatives of
analytic functions.
Here is the assignment for next time (not to hand in, but as usual
be prepared for a quiz).
Parts (b) and (c) of Problem 16 on page 74: namely,
compute
\[\int_C \,\frac{1}{z^2+1}\,dz\]
over the circle where \(|z+i|=1\) and over the circle where
\(|z|=2\), both oriented in the standard counterclockwise
direction.
Suggestion for part (c): Either decompose the integrand using the
method of partial fractions; or modify the curve \(C\) by
adding and subtracting a diameter of the circle to break the path
into two pieces, each surrounding only one of the points \(i\)
and \(-i\).
For the same two paths as in the preceding problem, evaluate
\[\int_C \,\frac{z}{z^2+1}\,dz.\]
Evaluate \(\displaystyle \int_{|z|=2}\,
\frac{z}{(z+1)^2}\,dz\), where the circular path is oriented
counterclockwise.
September 29
In class, we discussed two principles about invariance of line integrals of analytic functions under deformation of the path, and we saw the statement of Cauchy's integral formula that determines the value of an analytic function inside a curve through an integral over the curve.
Here is the assignment to hand in next class.
Problem 9 on page 73, which says the following (I am paraphrasing). Show that if \(C_1\) is the straight line segment joining the point \(0\) to the point \(2+i\), and \(C_2\) is the piecewise path that first joins \(0\) to \(2\) along the real axis and next joins \(2\) to \(2+i\) along a vertical line, then
\[\int_{C_1} \Re(z)\,dz \ne \int_{C_2} \Re(z)\,dz.\]
(In other words, the integral of \(\Re(z)\) along a path joining \(0\) to \(2+i\) is not invariant under deformation of the path.
The function \(\Re(z)\) is not analytic, so the conclusion does not violate the path-independence principle that we discussed in class.)
Suggestion: Compute the integrals by parametrizing the paths. The answer in the back of the book says that the values of the two integrals are \(2+i\) and \(2+2i\).
Problem 10(c) on page 73, which says to compute \(\int_C |z|\,dz\) when \(C\) is the circle where \(|z|=r\).
The answer in the back of the book is \(0\). But the function \(|z|\) is not analytic, so why doesn't this answer contradict Cauchy's integral theorem?
Problem 16(a) on page 74, which says to compute \(\displaystyle \int_C \,\frac{1}{z^2+1}\,dz\) when \(C\) is the circle where \(|z-i|=1\).
Suggestion: Factor the denominator as \((z+i)(z-i)\) and match the integral up with Cauchy's integral formula, taking \(z_0\) equal to \(i\).
We deduced Cauchy's integral theorem from Green's theorem, looked at some examples, and briefly reviewed for the exam to be given in class on Tuesday, September 27.
September 20
In class we discussed the notion of conformality, worked an example of a line integral, and discussed Green's theorem in the plane (which we will apply next time to analytic functions).
Here is the assignment for next time (not to hand in).
Problem 14 on page 44: Which part of the complex plane is stretched and which part shrunk under the following mappings?
\(w=z^2\)
\(w=z^2+z\)
\(w=1/z\)
Hint: At which points in the plane does the derivative have modulus greater than \(1\)? less than \(1\)?
Problem 10, parts (a) and (b), on page 73: Evaluate the integral \(\int_C |z|\,dz\), where \(C\) is
the segment of the real axis with initial point \(-1\) and final point \(1\),
the semicircle on which \(|z|=1\) and \(\Im z \ge 0\), again with initial point \(-1\) and
final point \(1\).
Answers from the back of the book: (a) \(1\) and (b) \(2\).
In preparation for the exam to be given on Tuesday, September 27, make a list of the main concepts covered so far.
September 15
In class, we discussed the assignment from last time, introduced some terminology (analytic functions, entire functions, and harmonic functions), and looked at the geometry of the mapping \(z\mapsto z^2\).
We ran out of time to take a quiz, so the assignment is to write solutions as a take-home quiz to hand in next class. Here are the three problems:
State the Cauchy–Riemann equations for a function \(f=u+iv\).
If \(f(z)=z+\overline{z}\), is \(f\) a continuous function? Explain why or why not.
If \(f(z)=z+\overline{z}\), is \(f\) an analytic function? Explain why or why not.
September 13
In class we discussed differentiability of functions of a complex variable and the Cauchy–Riemann equations.
Here is the assignment for next time (not to hand in, but be prepared for a quiz).
Problem 1 on page 43: Where is the function \(f(z)=z\,\Re(z)\) differentiable? How about the function \(f(z)=|z|\)?
Hint: You can use either the definition of the derivative as a limit, or the Cauchy–Riemann equations, or a combination of the two methods. You should find that the first function is differentiable only when \(z=0\), and the second function is nowhere differentiable.
Problem 3 on page 43: Prove that if \(f'(z)=0\) at every point of a domain \(G\), then \(f(z)\) is constant in \(G\).
Remark and hint: You know the corresponding statement for a function of a real variable from your first calculus class. But the proof uses the order relation on the real numbers (in the form of the mean-value theorem), so it is not obvious that the statement carries over to functions of a complex variable (where the order relation is lacking). The statement does, however, carry over to real-valued functions of two (or more) real variables: if \(u(x,y)\) is a real function for which both partial derivatives \(\partial u/\partial x\) and \(\partial u/\partial y\) are identically equal to \(0\), then \(u(x,y)\) is a constant function. You can use this theorem to solve the problem by translating the problem into a statement about the real part and the imaginary part of the complex-valued function \(f\) (by invoking the Cauchy–Riemann equations, for instance).
Problem 7 on page 43: Prove that the Cauchy–Riemann equations take the form
\[\frac{\partial u}{\partial r} = \frac{1}{r} \,\frac{\partial v}{\partial\theta}, \qquad
\frac{\partial v}{\partial r} = -\frac{1}{r}\, \frac{\partial u}{\partial \theta}
\]
in polar coordinates (\(x=r\cos\theta\), \(y=r\sin\theta\)).
Suggestion: Bring in the chain rule from two-dimensional real calculus. For instance,
\[\frac{\partial u}{\partial r} = \frac{\partial u}{\partial x} \, \frac{\partial x}{\partial r} + \frac{\partial u}{\partial y}\, \frac{\partial y} {\partial r}.
\]
Remark: The indicated equations break down when \(r=0\), because the transformation to polar coordinates breaks down when \(r=0\). Indeed, the angle \(\theta\) is undefined when \(r=0\).
September 8
In class, we discussed the three types of functions that will appear in this course: real-valued functions of a real variable, complex-valued functions of a complex variable, and complex-valued functions of a real variable (parametric curves). Also we discussed the notion of continuity of functions of a complex variable.
Then we split into groups to work on the following limits:
The assignment to hand in next time is to write up solutions to these three problems.
Remark: There is a subtlety in the third problem. A complex number has two square roots, which are negatives of each other; in language that we shall encounter later, the square-root function has two branches. The implicit assumption in the third problem is that the same choice of branch is made for both of the square roots. In other words, the arguments (angles) of \(\sqrt{n+2i}\) and \(\sqrt{n+i}\) should both be chosen to be close to \(0\) or both be chosen to be close to \(\pi\).
September 6
In class, we discussed the geometric interpretation of some functions of a complex variable, such as complex conjugation (reflection), multiplication by a positive real number (dilation), and multiplication by a complex number of modulus \(1\) (rotation). We discussed the solution of the homework problem about the equilateral triangle inscribed in the unit circle, and we looked at some examples of sequences and their limit points (limits of subsequences).
Here is the assignment about sequences to do for next class (not to hand in, but be prepared for a quiz).
Problem 11 on page 21, which says to show that the sequence with general term \(z_n=\frac{q^n}{1+q^{2n}}\) converges to \(0\) when \(q\) is a complex number of modulus less than \(1\) and also when \(q\) is a complex number of modulus greater than \(1\).
Hint: You can reduce the second case to the first case by dividing the numerator and denominator by \(q^{2n}\).
Remark: The problem is silent about the case when \(q\) is a complex number of modulus equal to \(1\), because nothing simple happens in that case. But what happens when \(q\) is a real number of modulus equal to \(1\)?
Problem 12(b) on page 21, which asks about the limit (if any) of the sequence with general term \(z_n=\sqrt[n]{n}+inq^n\), where \(q\) is a complex number of modulus less than \(1\).
Hint: Here \(\sqrt[n]{n}\) means the same thing as \(n^{1/n}\). If you do not remember from calculus about the limit of \(n^{1/n}\), think about what happens if you first take the logarithm and then take the limit. For the imaginary term, you could rewrite \(nq^n\) as \( (\sqrt[n]{n}\,q)^n\); does that help?
Problem 12(c) on page 21, which asks about the limit (if any) of the sequence with general term \(z_n=\sqrt[n]{a}+i\sin(1/n)\), where \(a\) is a positive real number.
Hint: If you are worried about the limit of \(\sqrt[n]{a}\), again try the trick of first taking the logarithm and then taking the limit. Remember to exponentiate at the end of the calculation to reverse the effect of taking the logarithm.
September 1
Here is the assignment to hand in next class: the following three problems.
Problem 3(d) on page 11, which says to calculate \(
\frac{(i+1)^9}{(1-i)^7}\).
Note added September 3: Instead of \((i+1)^9\),
the book has \((i+i)^9\) in the numerator, but that is a
typographical error. (If the author had really meant that, he would have
written \((2i)^9\).) Please do the problem as I have corrected it above.
My hint: Write the numerator and the denominator in trigonometric form and use De Moivre's theorem. WolframAlpha says that the answer is the real number \(2\). Do you agree?
Problem 13 on page 11, which says the following. Three points \(z_1\), \(z_2\), and \(z_3\) satisfy the conditions that \[z_1+z_2+z_3=0\] and \[|z_1|=|z_2|=|z_3|=1.\]
Show that the three points lie at the vertices of an equilateral triangle inscribed in the unit circle \(|z|=1\).
My hint: First try the special case when \(z_3=1\). If you can do that case, then work up to the general case. This is our first proof-type exercise. Your solution needs to be explained using sentences.
Problem 20(c) on page 12, which asks for all values of \(\sqrt[4]{-1}\), that is, of \((-1)^{1/4}\).
My hint: Use the trigonometric form. You should find four answers, which will be evenly spaced around the unit circle.
In class today, we reviewed the notions of real part, imaginary part, conjugate, modulus, argument, and De Moivre's theorem. We worked the second problem from the first assignment, and we used two methods to work out the cube roots of \(i\).
The first quiz was given. Here it is with solutions:
Solve \(\overline{z}=iz\), that is, determine all complex numbers whose conjugate equals \(i\) times the original number. Solution: If \(z=x+iy\), then the equation becomes \(x-iy=ix-y\). Matching the real parts shows that \(x=-y\), and matching the imaginary parts says again that \(x=-y\). Described geometrically, the solution is a line through the origin with slope equal to \(-1\). Described algebraically, the solution is all complex numbers of the form \(x-ix\), that is, all real multiples of the complex number \(1-i\).
Write the complex number \(1-i\) in trigonometric form, that is, in the form \(r(\cos(\theta)+i\sin(\theta))\). Solution: The modulus equals \(\sqrt{2}\) and the argument equals \(-\pi/4\), so the complex number can be written as
\[\sqrt{2}\,(\cos(-\pi/4)+i\sin(-\pi/4)).
\]
Solve \(z^2=i\), that is, determine all complex numbers whose square equals \(i\). First solution (using rectangular coordinates): If \(z=x+iy\), then \(z^2=x^2-y^2+2ixy\). For this expression to equal \(i\), the real part \(x^2-y^2\) must equal \(0\), and the imaginary part \(2xy\) must equal \(1\). The first condition means that \(x=\pm y\).
Consider the two cases separately. If \(x=+y\), then the second condition says that \(2x^2=1\), so \(x=\pm 1/\sqrt{2}\). Accordingly, there are two solutions for \(z\): namely, \((1+i)/\sqrt{2}\) and \(-(1+i)/\sqrt{2}\). On the other hand, if \(x=-y\), then the second condition says that \(-2x^2=1\), which is an impossible equation when \(x\) is a real number. Therefore the second case produces no additional solutions. Second solution (using polar coordinates): If \(z=r(\cos(\theta)+i\sin(\theta))\), then De Moivre's theorem implies that \(z^2=r^2(\cos(2\theta)+i\sin(2\theta))\). On the other hand, the trigonometric form of the complex number \(i\) is \(\cos(\pi/2)+i\sin(\pi/2)\). For the two expressions to match, \(r^2\) must equal 1, so \(r=1\), and the arguments must match. Therefore either \(2\theta=\pi/2\) or \(2\theta=2\pi+\pi/2=5\pi/2\). Accordingly, either \(\theta=\pi/4\) or \(\theta=5\pi/4\).
Thus the solutions for \(z\) are \(\cos(\pi/4)+i\sin(\pi/4)\) and \(\cos(5\pi/4)+i\sin(5\pi/4)\). Since \(\cos(\pi/4)=\sin(\pi/4)=1/\sqrt{2}\) and \(\cos(5\pi/4)=\sin(5\pi/4)=-1/\sqrt{2}\), the solutions can be written in standard form as \((1+i)/\sqrt{2}\) and \(-(1+i)/\sqrt{2}\), the same result as was obtained by the first method. Remarks: You could equally well rewrite \(1/\sqrt{2}\) as \(\sqrt{2}/2\). The preceding problem can be restated as asking for all possible square roots of the complex number \(i\). Every nonzero complex number has exactly two square roots, and they are negatives of each other.
August 30
Class began with an introduction to the subject of complex analysis. Then we worked Problem 2(a) on page 11 in groups and looked at parts (a) and (e) of Problem 5.
Here is the assignment due next class. (Since most of you are still trying to get the book, I have written out the statements of the problems.)
Do Problem 2(b) on page 11, which says to find all the complex numbers that are complex conjugates of their own cubes. (This problem is like the one we did in class, but with the cube instead of the square.)
Do Problem 5(f) on page 11, which says to determine the set of points \(z=x+iy\) in the complex plane satisfying the inequality
\[\left| \frac{z-1}{z+1}\right| \le 1.
\]
Here is some background for the next problem. You know that a point \((a,b)\) in the standard coordinate plane can be rewritten in polar form as \[(r\cos(\theta),r\sin(\theta)),\] where
\(r=\sqrt{a^2+b^2}\) (which is the modulus of the complex
number \(a+bi\)), and the angle \(\theta\) (called the
argument of the complex number \(a+bi\)) equals
\(\arctan(b/a)\). Thus a complex number \(a+bi\) can be written in
trigonometric form as \[r(\cos(\theta)+i\sin(\theta)).\] For example,
\(1+i=\sqrt{2}\,(\cos(\pi/4)+i\sin(\pi/4))\). Notice that the value of
the angle \(\theta\) is not uniquely determined: you can always add
\(2\pi\) to \(\theta\) without changing the values of
\(\cos(\theta)\) and \(\sin(\theta)\).
Now do parts (b) and (e) of Problem 6 on page 11, which ask for the trigonometric form of the complex numbers \(-1+i\) and \(1+\sqrt{3}\,i\).
Here is some background for the next problem. A formula called De Moivre's theorem says that
\[(\cos(\theta)+i\sin(\theta))^n = \cos(n\theta)+i\sin(n\theta)\]
for every positive integer \(n\). (This formula can be proved by
using trigonometric identities and mathematical induction.)
De Moivre's theorem is often an effective way to compute high
powers of complex numbers.
For example, suppose that you want to calculate
\((\sqrt{3}+i)^{407}\). Expanding directly by the binomial theorem
looks intractable. But if you first rewrite \(\sqrt{3}+i\) in
trigonometric form as \(2(\cos(\pi/6)+i\sin(\pi/6))\), then you can
apply De Moivre's theorem to deduce that
\[(\sqrt{3}+i)^{407} = 2^{407}(\cos(407\pi/6)+i\sin(407\pi/6)).
\]
The angle \(407\pi/6\) is equivalent to the angle \(-\pi/6\) (do you
see why?), so the answer reduces to
\[2^{407}(\cos(-\pi/6)+i\sin(-\pi/6)) = 2^{406}(\sqrt{3}-i).
\]
Now do Problem 15(a) on page 12, which asks for the value of
the quantity \((1+i)^{25}\).
I am not going to collect this initial assignment, but you should be
prepared for a quiz on Thursday (September 1) with problems of this nature.