Question

Notation discussion

Answer 1

\[\Huge
a^{1/n} =\sqrt[n]{a}e^{2\pi i/n} \\
\Huge
\frac{a^{1/n}}{\sqrt[n]{a}} =e^{2\pi i/n} \\
\Huge
[\frac{a^{1/n}}{\sqrt[n]{a}} ]^k=(e^{2\pi i/n} )^k\\

\Huge
\sqrt[n]{a}[\frac{a^{1/n}}{\sqrt[n]{a}} ]^k=\sqrt[n]{a}(e^{2\pi i/n} )^k\\

\]

Answer 2

\[\Large z = r \ e^{i \theta}\]

Answer 3

\[\Huge
e^i
\]

Answer 4

\[
(r,\theta)
\]

Answer 5

\[\Large e^{i (\theta + \phi)}=\cos(\theta+\phi)+i \sin(\theta+\phi)\]\[e^{i \theta}e^{i \phi}=( \cos \theta + i \sin \theta )(\cos \phi + i \sin \phi)\]\[\cos(\theta+\phi)+i stuff=\cos(\theta)\cos(\phi)-\sin (\theta)\sin(\phi) +i stuff\] \[\cos(\theta+\phi)=\cos(\theta)\cos(\phi)-\sin (\theta)\sin(\phi)\]

Answer 6

\[
(r,\theta)
\]

Answer 7

\[
\langle 1,\theta\rangle
\]

Answer 8

\[\Huge
x^{n^{(-1)}}
\]

Answer 9

\[\Large a^{-1}b^1c^id^{-i}\]

Answer 10

\(\color{blue}{\text{Originally Posted by}}\) @Kainui
\[\Large e^{i (\theta + \phi)}=\cos(\theta+\phi)+i \sin(\theta+\phi)\]\[e^{i \theta}e^{i \phi}=( \cos \theta + i \sin \theta )(\cos \phi + i \sin \phi)\]\[\cos(\theta+\phi)+i stuff=\cos(\theta)\cos(\phi)-\sin (\theta)\sin(\phi) +i stuff\] \[\cos(\theta+\phi)=\cos(\theta)\cos(\phi)-\sin (\theta)\sin(\phi)\]
\(\color{blue}{\text{End of Quote}}\)

Answer 11

\[
(r,\theta)
\]

Answer 12

\[
(r_1,\theta_1) + (0,\theta_2)
\]

Answer 13

\[
(r,\theta) = T(x,y)\\
(x,y) = T^{-1}(r,\theta)
\]

Answer 14

\[
\cos \equiv [T^{-1}(1,\theta)]_1
\]

Answer 15

\[
T(x,y) = (r,\theta)\implies T_1(x,y) = r
\]

Answer 16

\(\color{blue}{\text{Originally Posted by}}\) @wio
\[
T(x,y) = (r,\theta)\implies T_2(x,y) = \theta
\]
\(\color{blue}{\text{End of Quote}}\)

Answer 17

\[
\cos(\alpha) = T_1^{-1}(1,\alpha)
\]

Answer 18

\[
\sin(\alpha) = T_2^{-1}(1,\alpha)
\]

Answer 19

\[
\cos(\theta+\phi)=\cos(\theta)\cos(\phi)-\sin (\theta)\sin(\phi)

\] \[
(1,\theta) , (1,\phi)
\]

Answer 20

\[(a^2+b^2)(x^2+y^2)=(p^2+q^2)\]

Answer 21

\[(x \bar x)( y \bar y )=(xy)\bar{(xy)}\]

Answer 22

\[
a+bi = (a,b)
\]

Answer 23

\[
\bar z = (a,-b)
\]

Answer 24

\[
\overline{(a,b)} = (a,-b)
\]

Answer 25

\[z_1=(-a,b)\]\[z_2=(a,-b)\]

Answer 26

\[
[\cos(x) +i\sin(x)]^n = \cos(nx)+i\sin(nx)
\]

Answer 27

\[
e^{i} =\cos(x)+i\sin(x)
\]

Answer 28

\[
(\cos(x),\sin(x))
\]

Answer 29

\[
[\cos(x),\sin(x)]\cdot \overline{[\cos(y),\sin(y)]} = \cos(x)\cos(y)-\sin(x)\sin(y)
\]

Answer 30

\[
\overline{[a,b]} = [a,-b]
\]

Answer 31

\[
f(x) = [\cos(x),\sin(x)]
\]

Answer 32

\[
\cos(x+y) = f(x)\overline{f(y)} = \cos(x)\cos(y)-\sin(x)\sin(y)
\]

Answer 33

\[
\sin(x+y) = f(x)f(y) = \cos(x)\cos(y) + \sin(x)\sin(y)
\]

Answer 34

\[
\sin(x+y) =\ldots= \cos(x)\sin(y) +\cos(y) \sin(x)
\]

Answer 35

\[
[a,b]^T = [b,a]
\]

Answer 36

\[
\sin(x+y) = f(x)f^T(y) = \cos(x)\sin(y)+\sin(x)\cos(y)
\]

Answer 37

\[
[a,b,c]^T = [c,a,b]
\]

Answer 38

\[\Large \cos(\theta) = \frac{a \cdot b}{|a||b|}\]

Answer 39

\[
f(x) = [\cos(x),\sin(x)]
\]
\[
T^{-1}(1,x) =f(x)
\]

Answer 40

\[
f(\theta) = [\cos(\theta),\sin(\theta)]
\]
\[
T^{-1}(1,\theta) =f(\theta)
\]

Answer 41

\[\Large \cos \theta = \sqrt{ \frac{(a \cdot b)(a \cdot b)}{(a \cdot a)(b \cdot b)}}\]

Answer 42

\[
|a||b| =\sqrt{a\cdot a} \cdot \sqrt{b\cdot b}= \sqrt{a^2\cdot b^2}
\]

Answer 43

\[\Large \cos \theta
= \frac{a\cdot b}{\sqrt{(a \cdot a)(b \cdot b)}}\]

Answer 44

\[
a+bi \iff (x,y)
\]

Answer 45

\[
e^i = \cos(\theta)+i\sin(\theta)
\]

Answer 46

\[
\overline{[x,y]} = [x,-y]
\]

Answer 47

\[
[x,y]_1 = [-x,y]
\]

Answer 48

\[
[x,y]_2 = [x,-y]
\]

Answer 49

\[
T([x,y]) = [r,\theta]
\]

Answer 50

\[
\overline z = [x,y]_2
\]

Answer 51

\[
z_1+z_2 = (a_1,b_1)+(a_2,b_2) = (a_1+a_2,b_1+b_2)
\]

Answer 52

\[
z\times z'= (a+bi)(a'+b'i) = aa'+ab'i+a'bi-bb'
\]

Answer 53

\[
(aa'-bb',ab'+a'b)
\]

Answer 54

\[
(a,b)\times (a',b') = (aa'-bb',ab'+a'b)
\]

Answer 55

\[(ai+bj)(a'i+b'j)=aa'ii+bb'jj+ba'ji+a'bij\]

Answer 56

ii=1
ij=-ji

Answer 57

\[(aa'+bb')+(a'b-b'a)ij = v \cdot v + v \times v\]

Answer 58

\[
[x,y]_1 = [-x,y]
\]

Answer 59

\[
[x,y]^1 = [x,y]\\
[x,y]^2 = [y,x]
\]

Answer 60

\[
\sigma_2[x,y]
\]

Answer 61

\[
\varsigma_2[x,y] = [x,-y]
\]

Answer 62

\[
[x,y]\times [x',y'] = \ldots = [xx'-yy',x'y+xy']
\]

Answer 63

\[
[x,y]\times [x',y'] = z \times z' = [z \cdot \varsigma_2(z'), z\cdot \sigma_2(z')]
\]

Answer 64

\[\overline{(a+bi)}(a'+b'i)=(aa'+bb')+(ab'-a'b)i\]

Answer 65

\[
z\cdot z' = (a+bi)(a'+b'i)
\]

Answer 66

\[
[\cos (\theta)+i\sin(\theta)]^n = \cos(n\theta)+i\sin(\theta)
\]

Answer 67

\[
f(\theta) = [\cos(\theta),\sin(\theta)]
\]
\[
T^{-1}(1,\theta) =f(\theta)
\]
\[
[f(\theta)]^n = f(n\theta)
\]

Answer 68

\[
\exp(i\theta) = \cos(\theta)+i\sin(\theta)
\]

Answer 69

\[
z=r[\cos(\theta )+i\sin(\theta)]
\]

Answer 70

\[
f(\theta)f(\phi) = f(\phi+\theta)
\]

Answer 71

\[
f(\theta\phi) = f(\theta)+f(\phi)
\]

Answer 72

\[\large e^x=\sum_{n=0}^\infty \frac{x^n}{n!}\]

Answer 73

\[\large \cos(x)=\sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!}\]

Answer 74

\[
[T^{-1}(r,\theta)][T^{-1}(\rho , \phi)] = T^{-1}(r\rho,\theta+\phi)
\]

Answer 75

\[
[x,y]
\]

Answer 76

\[
[x,y][T^{-1}(1,\phi)] = [x',y']
\]

Answer 77

\[
[x,y][\cos(\phi),\sin(\phi)]=[x',y']
\]

Answer 78

\(\color{blue}{\text{Originally Posted by}}\) @wio
\[
[x,y]\times [x',y'] = \ldots = [xx'-yy',x'y+xy']
\]
\(\color{blue}{\text{End of Quote}}\)

Answer 79

\[
R_{2\times 2}(\theta)
\]

Answer 80

\[
z=[a,b]\\
[z,z'] = \begin{bmatrix}
a&a' \\
b&b'
\end{bmatrix}
\]

Answer 81

\[
z+z'j
\]

Answer 82

\[
z=a+bi = ai+bj
\]

Answer 83

\[
i^2=1,j^2=-1
\]

Answer 84

\[a+bij\]

Answer 85

\[(a+bij)^~=a+bji=a-bij\]

Answer 86

\[(ij)(ij)=-1\]

Answer 87

\[
[\cos(\theta),\sin(\theta)]
\]

Answer 88

\[
sin(x+y)=f(x)fT(y)=cos(x)sin(y)+sin(x)cos(y)
\]

\[
s=\sin(x)
\]
\[
s(x+y) = s'(x)s(y) + s(x)s'(y)
\]

Answer 89

\[[s(x)s(y)]'\]

Answer 90

\[
s(x+y) =[s(x)s(y)]'=s'(x)s(y)+s(x)s'(y)
\]