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这是阿拉巴马大学复分析课程的代写成功案例。
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.
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Math113 Complex Analysis HELP（EXAM HELP， ONLINE TUTOR）
1.1.3. Add the following columns to the truth table in (5): $P$ and $\operatorname{not} Q,(\operatorname{not} P)$ and (not $Q)$, $(\operatorname{not} P)$ or $(\operatorname{not} Q), P \Rightarrow \operatorname{not} Q,(\operatorname{not} P) \Rightarrow$ not $Q$. Are any of the new columns negations of the columns in the truth table (5) or of each other?
1.1.4. Suppose that $P \Rightarrow Q$ is true and $Q$ is false. Prove that $P$ is false.
1.1.5. Prove that $P \Rightarrow Q$ is equivalent to ( $\operatorname{not} Q) \Rightarrow$ ( not $P)$.
1.1.6. Simplify the following statements:
i) $(P$ and $P)$ or $P$.
ii) $P \Rightarrow P$.
iii) $(P$ and $Q)$ or $(P$ or $Q)$.
1.1.7. Prove with truth tables that the following statements are true.
i) $(P \Leftrightarrow Q) \Leftrightarrow[(P \Rightarrow Q)$ and $(Q \Rightarrow P)]$.
ii) $(P \Rightarrow Q) \Leftrightarrow(Q$ or $\operatorname{not} P)$.
iii) $(P$ and $Q) \Leftrightarrow[P$ and $(P \Rightarrow Q)]$.
iv) $[P \Rightarrow(Q$ or $R)] \Leftrightarrow[(P$ and $\operatorname{not} Q) \Rightarrow R]$.
1.1.13. (The quadratic formula) Let $a, b, c$ be real numbers with $a$ non-zero. Prove (with algebra) that all solutions $x$ of the quadratic equation $a x^2+b x+c=0$ are of the form
$$
x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}
$$
1.1.14. Prove that $\sqrt{3}$ is not a rational number. (Remark: It is harder to prove that $\pi$ and $e$ are not rational.)
1.1.15. For which integers $n$ is $\sqrt{n}$ not a rational number?
1.2.5. (Contrast with the switching of quantifiers in the previous two exercises.) Explain why the following two statements do not have the same truth values:
i) For every $x>0$ there exists $y>0$ such that $x y=1$.
ii) There exists $y>0$ such that for every $x>0, x y=1$.
1.2.6. Rewrite the following statements using quantifiers:
i) 7 is prime.
ii) There are infinitely many prime numbers.
iii) Everybody loves Raymond.
iv) Spring break is always in March.
1.2.7. Let ” $x L y$ ” represent the statement that $x$ loves $y$. Rewrite the following statements symbolically: “Everybody loves somebody,” “Somebody loves everybody,” “Somebody is loved by everybody,” “Everybody is loved by somebody.” At least one statement should be of the form ” $\forall x \exists y, x L y$ “. Compare its truth value with that of ” $\exists x \forall y, x L y$ “.
1.2.8. Find a property $P$ of real numbers $x, y, z$ such that ” $\forall y \exists x \forall z, P(x, y, z)$ ” and $” \forall z \exists x \forall y, P(x, y, z) “$ have different truth values.
1.2.9. Suppose that it is true that there exists $x$ of some kind with property $P$. Can we conclude that all $x$ of that kind have property $P$ ? A mathematician and a few other jokers are on a train and see a cow through the window. One of them generalizes: “All cows in this state are brown,” but the mathematician corrects: “This state has a cow whose one side is brown.”
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