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Euler form of complex number - (Concept)
The polar form of complex number z = r (cos ? + i sin ?)
In Euler form (cos ? + i sin ?) part of the polar form of complex numbers is represented by eiΘ. So, z = r (cos ? + i sin ?) is written as r.eiΘ in Euler's Form
We know the expansion of ex is
The expansion of $e^x$ is
$
\mathrm{e}^{\mathrm{x}}=1+\frac{\mathrm{x}}{1!}+\frac{\mathrm{x}^2}{2!}+\frac{\mathrm{x}^3}{3!}+\ldots
$
Replacing x with ix
$
\begin{aligned}
& e^{i x}=1+\frac{i x}{1!}+\frac{(i x)^2}{2!}+\frac{(i x)^3}{3!}+\frac{(i x)^4}{4!}+\ldots \\
& e^{i x}=1+\frac{i x}{1!}-\frac{x^2}{2!}-\frac{i x^3}{3!}+\frac{x^4}{4!}+\ldots+
\end{aligned}
$
rearranging the terms, we have
$
e^{i x}=\left(1-\frac{x^2}{2!}+\frac{x^4}{4!}\right)+i\left(x-\frac{x^3}{3!}+\frac{x^5}{5!}\right)
$
We notice that first bracket is the expansion of $\sin x$ and 2nd bracket is the expansion of $\cos x$, so we have $e^{i x}=\sin x+i \cos x$
So, eix = cosx + isinx and
e-ix = cosx - isinx
Euler forms make algebra very simple for complex numbers in cases where multiplication, division or powers of complex numbers are involved. Any complex number can be expressed as
$
\begin{aligned}
& \mathrm{z}=\mathrm{x}+\mathrm{iy} \\
& \mathrm{z}=\mid \mathrm{z}(\cos \theta+\mathrm{i} \sin \theta) \\
& \mathrm{z}=|\mathrm{z}| \mathrm{e}^{i \theta}
\end{aligned}
$
(Cartesian form)
(Polar form)
(Euler's form)
Application of Euler form:
1. Multiplication of two complex numbers:
Let $z=|z| e^{i \theta_1}$
And $\mathrm{w}=|\mathrm{w}| \mathrm{e}^{\mathrm{i} \theta_2}$
Multiplying these two number
$
\begin{aligned}
& \mathrm{z} \cdot \mathrm{w}=|\mathrm{z}| \mathrm{e}^{\mathrm{i} \theta_1} \cdot|\mathrm{w}| \mathrm{e}^{\mathrm{i} \theta_2} \\
& =|\mathrm{z}| \cdot|\mathrm{w}| \mathrm{e}^{\mathrm{i}\left(\theta_1+\theta_2\right)}
\end{aligned}
$
2. Division also can be done in the same way,
$\mathrm{z}=|\mathrm{z}| \mathrm{e}^{\mathrm{i} \theta_1}$ and $\mathrm{w}=|\mathrm{w}| \mathrm{e}^{\mathrm{i} \theta_2}$ be two complex number
$
\therefore \frac{\mathrm{z}}{\mathrm{w}}=\frac{|\mathrm{z}|}{|\mathrm{w}|} \mathrm{e}^{\mathrm{i}\left(\theta_1-\theta_2\right)}
$
3. The logarithm of Complex Number
$
\begin{aligned}
& \mathrm{z}=|\mathrm{z}| \mathrm{e}^{\mathrm{i} \theta} \\
& \log _{\mathrm{e}}(\mathrm{z})=\log _{\mathrm{e}}\left(|z| \mathrm{e}^{\mathrm{i} \theta}\right) \\
& \left.\log _{\mathrm{e}}(\mathrm{z})=\log _{\mathrm{e}}|\mathrm{z}|\right)+\log _{\mathrm{e}}\left(\mathrm{e}^{\mathrm{i} \theta}\right) \\
& \log _{\mathrm{e}}(\mathrm{z})=\log _{\mathrm{e}}(|z|)+\mathrm{i} \arg (\mathrm{z})
\end{aligned}
$
| Exam | Chapter |
| JEE MAIN | Complex numbers and quadratic equations |
If $\alpha$ and $\beta$ are the roots of the equation $\frac{1}{i Z}-i Z=2(\sin \theta-i \cos \theta)$ where $0<\theta<\pi$ and $i=\sqrt{-1}$, and z is complex number, then the value of $|\alpha-i|+|\beta-i|$ is $2\sqrt n$. Find the value of $n$.
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, its Euler form is?
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If z is a non-real complex number, then the minimum value of $\frac{Im \: z^{5}}{\left ( Im\: z \right )^{5}}$
| A. |
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If z is a complex number of unit modulus and argument ,then arg
equals:
| A. |
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If and
are two complex numbers such that
and
then :
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The value of
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Let If
and
be the greatest integral part of
Then
is equal to
| A. |
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| D. |
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Euler's form of is
| A. |
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Euler's form of will be
| A. |
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If $z$ and $\omega$ are two nonzero complex numbers such that $:|z \omega|=1$, and $\operatorname{Arg}(z)-\operatorname{Arg}(\omega)=\pi / 2$, then $\overline{\mathbf{z}} \omega$ is equal to
| A. |
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| D. |
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Real part of is equal to:
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z and w are two non-zero complex number such that | z | = | w | and Arg z + Arg w = then z equals
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| C. |
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| D. |
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$
\text { What is the euler form of complex number } z=2 \sqrt{3}-2 i?
$
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| B. |
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| C. |
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| D. |
|
If $e^{i \alpha}=\cos \alpha+i \sin \alpha$, then for the $\triangle A B C,$ the value of $ e^{i A} \times e^{i B} \times e^{i C}$ is:
| A. |
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| B. |
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| D. |
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If $z=\cos \theta + i \sin \theta,$ then:
| A. |
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| C. |
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| D. |
|
Let $z=18+26 i$ and $z_1=a+i b(a, b \in R)$ be the cube root of $z$ having the least positive argument. Find the value of $a b(a+b)$.
| A. |
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| B. |
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| D. |
|
Let $z$ be a complex number such that $|z+2|=1$ and $\operatorname{lm}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $|\operatorname{Re}(\overline{z+2})|$ is:
| A. |
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| B. |
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| C. |
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| D. |
|
The Complex number
$
z=\frac{\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)^3 \cdot(\sqrt{3}-i)^4}{i^5} \text { equals }
$
| A. |
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| B. |
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| C. |
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| D. |
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For any integer $k, \alpha_k=\cos \frac{k \pi}{9}+i \sin \frac{k \pi}{9}$, and $i=\sqrt{-1}$. Then the value of $\frac{\sum_{k=1}^{15}\left|\alpha_k-\alpha_{k-1}\right|}{\sum_{k=1}^3\left|\alpha_{3 k-1}-\alpha_{3 k-2}\right|}$ is:
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| B. |
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| D. |
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The real part of $\left[1+\cos \left(\frac{\pi}{5}\right)+i \sin \left(\frac{\pi}{5}\right)\right]^{-1}$ is:
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Let r be a real number and n ∈ N be such that the polynomial 2x2 + 2x + 1 divides the polynomial (x + 1)n – r. Then (n, r) can be.
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Let $\mathrm{z}_1, \mathrm{z}_2$ and $\mathrm{z}_3$ be three complex numbers on the circle $|z|=1$ with $\arg \left(z_1\right)=\frac{-\pi}{4}, \arg \left(z_2\right)=0$ and $\arg \left(z_3\right)=\frac{\pi}{4}$. If $\left|z_1 \bar{z_2}+z_2 \bar{z}_3+z_3 \bar{z_1}\right|^2=\alpha+\beta \sqrt{2}, \alpha, \beta \in Z$, then the value of $\alpha^2+\beta^2$ is :
| A. |
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| D. |
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The number of complex numbers $z$, satisfying $|z|=1$ and $\left|\frac{\mathrm{z}}{\overline{\mathrm{z}}}+\frac{\overline{\mathrm{z}}}{\mathrm{z}}\right|=1$, is:
| A. |
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| B. |
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| C. |
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| D. |
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For any real number $r$. let $A_r=\left\{e^{i \pi r n}: n\right.$ is a natural number $\}$ be a set of a complex number. Then
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