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Math113课程简介
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We have designed this course on Analysis I: Complex Function Theory for those who are studying at Harvard University and want to consolidate their understanding and learn from experts and professors, at any time, and from anywhere.
This personalized course is stocked with video tutorials, practice questions, study guides, and 3 personalized tutor responses to ensure that you fully understand the material in all the subjects.
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Prerequisites
COURSE DESCRIPTION
This course is a continuation of Math 112. It includes the study of trigonometric and inverse trigonometric functions, as well as extensive work with trigonometric identities and equations. Other topics contained in this course are: polar coordinates, complex numbers, DeMoivre’s Theorems, vectors, and conic sections.
The course presents the essential characteristics and basic processes of inquiry and analysis in the discipline.
The course encourages the development of critical thinking skills and requires students to analyze, synthesize, and evaluate
knowledge.
The course considers its subjects in relation to other disciplines and to the human condition.
The course is not limited to majors in any discipline.
The course does not focus on professional skills.
LEARNING OUTCOMES FOR THIS COURSE
Math113 Complex Analysis HELP（EXAM HELP， ONLINE TUTOR）
Collaboration: On the homework sets, collaboration is not only allowed but encouraged. However, you must write up and understand your own individual homework solutions, and you may not share written solutions. If you learn how to solve a problem by talking to a classmate, CA, or looking it up in a book, you should cite the source in your homework write-up, just as you would reference your sources in a literature or history class. Show all your working, and write up your solutions as neatly as possible.
Solve the following equations in complex numbers, and write the answer in polar form and rectangular form respectively.
$z^2-i=0$
$$
z^2-i=0
$$
Solution. Write $z=r e^{i \theta}$. Then the equation reads $r^2 e^{2 i \theta}=e^{i \pi(2 n+1) / 2}$. The restriction $0 \leq \theta \leq 2 \pi$ gives the solutions $z=e^{i \pi / 4}$ and $z=e^{5 i \pi / 2}$ corresponding to the rectangular forms $\frac{1}{\sqrt{2}}(1+i)$ and $-\frac{1}{\sqrt{2}}(1+i)$, respectively.
(b) $z^7+z^6+z^5+z^4+z^3+z^2+z+1=0$.
$$
z^7+z^6+z^5+z^4+z^3+z^2+z+1=0
$$
Solution. Note that $(z-1)\left(z^7+z^6+z^5+z^4+z^3+z^2+z+1\right)=z^8-1$, so the solutions to our equation correspond to the seven non-trivial $8^{\text {th }}$ roots of unity. In polar form, they are $e^{i \pi k / 4}, k=1, \ldots, 7$. In rectangular coordinates they are $\frac{1}{\sqrt{2}}(1+i), i, \frac{1}{\sqrt{2}}(-1+i)$, $-1, \frac{1}{-\sqrt{2}}(1+i),-i$ and $\frac{1}{\sqrt{2}}(1-i)$, respectively.
$e^{2 z}+2 e^z=-2$. (Omit writing $z$ in polar form for this question.)
$$
e^{2 z}+2 e^z=-2
$$
Solution. We treat the above equation as a quadratic in $e^z$. By use of the quadratic formula we compute $e^z=-1 \pm i$. Restricting to the principal branch of the logarithm (i.e., where the argument is in the range $(-\pi, \pi))$ we obtain $z=\log (\sqrt{2})+3 \pi i / 4$ and $z=\log (\sqrt{2})-3 \pi i / 4$
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