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Complex Variables With Applications

S. Ponnusamy, Herb Silverman

Chapter 1

Algebraic and Geometric Preliminaries - all with Video Answers

Educators


Section 1

The Complex Field

07:39

Problem 1

Show that the set of real numbers of the form $a+b \sqrt{2}$, where $a$ and $b$ are rational, is an ordered field.

Mengchun Cai
Mengchun Cai
Numerade Educator
01:43

Problem 2

$$
\begin{aligned}
&\text { If } a \text { and } b \text { are elements in a field, show that } a b=0 \text { if and only if either }\\
&a=0 \text { or } b=0
\end{aligned}
$$

Manisha Sarker
Manisha Sarker
Numerade Educator
01:14

Problem 3

Suppose $a$ and $b$ are elements in an ordered field, with $a<b$. Show that there are infinitely many elements between $a$ and $b$.

Doruk Isik
Doruk Isik
Numerade Educator
02:33

Problem 4

Find the values of
(a) $(-2,3)(4,-1)$
(b) $(1+2 i)\{3(2+i)-2(3+6 i)\}$
(c) $(1+i)^{3}$
(d) $(1+i)^{4}$
(e) $(1+i)^{n}-(1-i)^{n}$.

Aman Gupta
Aman Gupta
Numerade Educator
03:21

Problem 5

Express the following in the form $x+i y$ :
(a) $(1+i)^{-5}$
(b) $(3-2 i) /(1-i)$
(c) $e^{i \pi / 2}+\sqrt{2} e^{i \pi / 4}$
(d) $(1+i) e^{i \pi / 6}$
(e) $\frac{a+i b}{a-i b}-\frac{a-i b}{a+i b}$
(f) $\frac{3+5 i}{7+i}+\frac{1+i}{4+3 i}$
(g) $(2+i)^{2}+(2-i)^{2}$
(h) $\frac{(4+3 i) \sqrt{3+4 i}}{3+i}$
(i) $\frac{\left(a i^{40}-i^{17}\right)}{-1+i},(a-$ real $)$
(j) $(-1+i \sqrt{3})^{60}$
(k) $\frac{\sqrt{1+a^{2}}+i a}{a-i \sqrt{1+a^{2}}},(a-$ real $)$
(l) $\frac{(\sqrt{3}-i)^{2}(1+i)^{5}}{(\sqrt{3}+i)^{4}}$.

Linda Hand
Linda Hand
Numerade Educator
01:39

Problem 6

Show that
$$
\left(\frac{-1 \pm \sqrt{3}}{2}\right)^{3}=1 \text { and }\left(\frac{\pm 1 \pm i \sqrt{3}}{2}\right)^{6}=1
$$
for all combinations of signs.

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
04:39

Problem 7

For any integers $k$ and $n$, show that $i^{n}=i^{n+4 k} .$ How many distinct values can be assumed by $i^{n} ?$

Willis James
Willis James
Numerade Educator