Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

learn more… | top users | synonyms

0
votes
1answer
12 views

Derivatives of real part of function

My physics textbook gives me the complex form equation of simple harmonic motion as: $$z = Ae^{i(\omega _{o}t+\phi )}$$ and then defines $$ x = Re (z) $$ From there they argue that $$ \frac{\partial ...
1
vote
1answer
22 views

Explanation of two argument variant for arctan

Can someone please explain why $$\tan^{-1}\left(\frac{y}{x}\right)$$ has the additional conditions based on what the value of x and y are? I'm most specifically interested in the second equation: ...
0
votes
0answers
11 views

Simplifying complex exponential

Two simplifications from my book that I don't understand, first: 5exp(-j1.571) = - j5 Why does the real part get dropped off? Also: exp(j3.785) = -exp(0.643) Is there a way to directly covert ...
1
vote
2answers
37 views

Using Eulers formula

I am trying to figure out how \begin{equation*} e^{i(-1+i\sqrt{3})}=e^{-\sqrt{3}} (cos(1)-i sin(1))?? \end{equation*} I know that Euler's formula states that \begin{equation*} e^{ix} = \cos(x) + i ...
0
votes
0answers
18 views

complex integral evaluation strategy

I am trying to evaluate $$\int \frac{z}{z+2} dz$$ counter clockwise on the circle $|z|=1$ what is the general strategy? The denominator has a pole but it is outside the circle. By the Cauchy theorem I ...
0
votes
1answer
21 views

Evaluating complex integral on circle

I am trying to evaluate the integral $$\int \frac{2z-1}{z(z-1)} dz$$ counter clockwise around the circle $$|z|=2$$ First we apply partial fraction decomposition to get $$\int \frac{1}{z}+\int ...
0
votes
2answers
14 views

Complex number (Rhombus)

Given that $z_1=1+2i$ and $z_2=\frac{3}{5}+\frac{4}{5}i$, write $z_1z_2$ and $\frac{z_1}{z_2}$ in the form $p+iq$, where $p$ and $q \in R$. In an Argand diagram, the origin O and the points ...
1
vote
1answer
25 views

Sum of complex series

After stating the sum I wrote z in polar form and then proceeded to calculate the real part of the sum I stated in the first part. However the working got tedious very soon and I was not able to ...
-1
votes
1answer
20 views

Argument of Complex Number (Am I wrong?)

I'm given $z=-2+\sqrt{3}i$. So I worked out the argument of $arg(z)=\tan^{-1}(\frac{\sqrt{3}}{-2})$. I got the answer $2.256$rad. But the given answer is $2.45$rad. Am I wrong?
0
votes
0answers
13 views

Integrating complex functions over the unit circle

I am trying to evaluate $$\int_c \bar z dz$$ where the contour is the unit circle. I know the limits of $\theta$ is $0 \to 2\pi$ How do I get to the answer of $2\pi i$?
4
votes
3answers
66 views

Finding local max of analytic function

Given a function $f=z^2+iz+3-i$. I need to find the the maximum of $|f(z)|$ in the domain $|z|\leq 1$ I know that the maximimum should be on $|z|=1$ but when I tried to put $z=e^{i\theta} $ in the ...
1
vote
2answers
17 views

Simplifying Complex numbers

Help me simplify this complex number: Hints are welcome, so that I can see how to move on $$\left(\frac{1+6i}{\sqrt{76}e^{\frac{1}{2}\pi i}}\right)^{2i}$$
0
votes
3answers
39 views

Complex number $\tan \alpha+i$

Given that $z=\tan \alpha+i$, where $0<\alpha<\frac{1}{2}\pi$ Find $\left |z \right |$. I've never seen this kind of example in my book. Can anyone guide me? Thanks a lot. How to find $arg ...
3
votes
2answers
174 views

Square roots of Complex Number. [duplicate]

Calculate, in the form $a+ib$, where $a,b\in \Bbb R$, the square roots of $16-30i$. My attempt with $(a+ib)^2 =16-30i$ makes me get $a^2+b^2=16$ and $2ab=−30$. Is this correct?
1
vote
1answer
32 views

Find $\int_c \bar z$ along the parabola $y=x^2$ from $(0,0)$ to $(1,1)$

I know $\bar z=x-iy$ So we have $$\int_c x-iy \,dz$$ when split up gives us $$\int^1_0 x \, dx-i\int^1_0 x^2 \cdot 2x \, dx$$ and then I integrate as usual as usual and I get the result ...
0
votes
2answers
21 views

equation of a line into the complex form

So if i am given an equation of a line in complex form for example $Re|(1+i)z| = 0$, I could turn this into its real counter part on the x-y plane and graph it. Is there a way to go in the other ...
0
votes
0answers
26 views

complex variable inequality

Let $B$ and $C$ be nonegative real numbers and $A$ a complex number. Suppose that $$ 0\leq B-2Re(\overline{\lambda}A) + |\lambda|^2 C \ \forall \ \lambda \in \mathbb{C} $$ Conclude that $|A|^2 \leq ...
0
votes
1answer
25 views

How to find complex coordinates of a square?

If one coordinate is given by: $z_{1}=\frac{3}{2}+\frac{3}{2}i$ and $Re(z_{2})=6,Re(z_{4})=1$. How to find $z_{2},z_{3},z_{4}$ so that $z_{1}z_{2}z_{3}z_{4}$ forms a square in the first quadrant? ...
1
vote
0answers
32 views

Complex numbers and simple argument question

Yesterday, i encountered a question: $z=a+bi$ $Arg(z-\overline z + 4) = {4\pi \over 3}$ $b=?$ I solved the question using basic method: $$\overline z = a-bi$$ $$ w = z - \overline z + ...
0
votes
0answers
25 views

How is the second part of a dual number called?

A complex number $a + bi$ has a real part $a$ and an imaginary part $b$. But, what about dual numbers $u + v\epsilon $? I have seen the non-real part $v$ been called the infinitesimal part. Is this a ...
2
votes
2answers
44 views

Argument of complex number $(\tan \theta)$

I'm given $-2+2\sqrt{3}i$. The question asks me to find the argument. My attempt, $\tan \theta=\frac{2\sqrt{3}}{2}$ So $\theta=\frac{\pi}{3}$. But the given answer is $\frac{2\pi}{3}$. Why?
1
vote
3answers
30 views

Understanding quotients of complex numbers

I am reading an old complex variables textbook which states: Given $z = a + bi$, $z_1 = a_1 + b_1i$, and $z_2 = a_z + b_2i \neq 0$, we have $z = \dfrac{z_1}{z_2} = \dfrac{a_1a_2 + b_1b_2}{a_2^2 + ...
3
votes
7answers
95 views

Show that $\cos(6x)= 32\cos^6x -48\cos^4x +18\cos^2x -1$

After writing down $\cos6x$= $Re (\cos x + i\sin x)^6$, I used the binomial theorem to expand the expression. Very soon it got really tedious and after trying $5$ times, fruitlessly, to arrive at the ...
0
votes
1answer
19 views

Derivative of a complex conjugate

I anticipate that this is a stupid question, but suppose $c \in C$. What is $\frac{\partial c^{*}}{\partial c}$? I've been trying and failing for about an hour to figure it out from the definition of ...
4
votes
1answer
76 views
+100

Mandelbrot set of $c \cdot \cos(z)$

I'm given a task to write a program, that determines if a given point $c \in \mathbb{C}$ is in the Mandelbrot set of the function $$f_c(z) = c \cdot \cos (z)$$ That is if the set $\{z_n = f_c^n (0) : ...
1
vote
2answers
32 views

Inequality $\left|z+w\right|\geq\left||z|-|w|\right|$ when $|w|\leq A|z|$.

Let $z,w\in\mathbb C$ and $|w|\leq A|z|$ for $A>0$. I want estimate from below $|z+w|$. I proceeded as follows. Since $$\left|z+w\right|\geq\left||z|-|w|\right|$$ and $-|w|\geq -A|z|$, I write ...
3
votes
1answer
16 views

$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$ onto upper half space $\{\,z\in \mathbb C : \operatorname{Im}(z)>0 \,\}$

I am search an one to one mapping that maps the domain $$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$$ onto upper half space $$\{\,z\in \mathbb C : \operatorname{Im}(z)>0\, ...
1
vote
1answer
16 views

Contour integral of non analytic exponential function

The value of the integration of the function $f(z)$ over the circle of radius 3 centered at $z=1$, where $f(z) = e^{\frac{-1}{(z-1)^2}}$ this function has a pole at $z=1$ of $2$nd order. I don't ...
2
votes
2answers
27 views

Complex Numbers (Geometric Representations)

What is the geometrical interpretation of this operation: Multiplication by $\frac{\left(1-i\right)}{\sqrt{2}}$ Attempt: multiplication by −i = rotate by −π/2
2
votes
7answers
129 views

Why is the angle of $i^2 = \pi$?

On the complex plane , the angle of $i = \pi / 2$ and the angle of $i^2 = \pi$ . I understand that by definition $i^2 = -1$ but do not understand how to arrive at angle $\pi$ from $\pi / 2$ when ...
3
votes
3answers
40 views

Determine a complex conjugate to $u(x,y)=x^3y-xy^3$

I know $\frac{\partial^2 u}{\partial x^2}=6xy$ and $\frac{\partial ^2 u}{\partial y^2} =-6xy$ and adding these together I get 0 which tells me they are harmonic functions. To determine the harmonic ...
3
votes
2answers
113 views

Guessing the other root to a quadratic equation

I just attempted to do the question below, but it seems that even after seeing the answer I'm not sure I understand the motivation for the solution. Let $\alpha ...
1
vote
1answer
45 views

Changing argument into complex in the integral of Bessel multiplied by cosine

I got a problem solving the equation below: $$ \int_0^a J_0\left(b\sqrt{a^2-x^2}\right)\cosh(cx) dx$$ where $J_0$ is the zeroth order of Bessel function of the first kind. I found the integral ...
-1
votes
0answers
22 views

Sketching the image of a circle under a complex polynomial

I want to sketch $w = z^3 + z^2 - iz + 1$ for $|z| = 2$. Finding the relation between $U(x,y)$ and $V(x,y)$ is my main question. I found $V^2 = (U - (x^3 + x^2 + 1))^3$ but I don't know how to use ...
1
vote
7answers
77 views

Solving $z+i\overline{z}=iz-\overline{z}$

I want to solve $z+i\overline{z}=iz-\overline{z}$ ($\overline{z}$ is the complex conjugate). I have solved it setting $z=a+bi$. But can one solve without writing it $z$ a certain form, factorization ...
0
votes
3answers
40 views

Solution of an equation with complex numbers [on hold]

Knowing that $2+i$ is a solution of $z^3 - 11z + 20 = 0 $ Calculate the other solutions
0
votes
3answers
70 views

prove that $f(z)+f(iz)=0$ please

When $f(z).f(iz)=z^2\space \forall z \in \mathbb{C}$ How to prove that $f(z)+f(iz)=0 \ \ \space \forall z \in \mathbb{C}$ I try Let $f(z)+f(iz)=M$ $ f(z)=\frac {z^2}{f(iz)}$ ...
0
votes
2answers
31 views

Problem about complex number

Find all values of $(-1)^{1/3}$ I used the identity's and such and got a part where I got $e^{1/3\log(-1)}$, and I'm not sure how to do the next step and get to the answer. Can anyone send in the ...
1
vote
1answer
21 views

Find the values of $a,b,c$ of the complex function $f(x)= (ax+b)/(x+c)$

The task is to find the values of $a$,$b$, and $c$ of the complex function $f(x)=\frac{ax+b}{x+c}$ where $a,b,c \in \mathbb{R}$. It is given that $f(2i)=-2i$ and $f(1+3i)=1-3i$. I tried to make an ...
1
vote
1answer
19 views

Showing the limit does not exist

I am trying to show $\lim_{z \to 0} f(z)$ does not exist where $f(z)=\frac{xy}{2x^2+3y^2} +ix^2$. I am to show the limit does not exist by taking the limit along the straight line $y=mx$ where m is a ...
4
votes
3answers
44 views

Express $w=f(z)=\frac{1}{(1-z)^2}$ in the form $w=u(x,y)+iv(x,y)$

I start by writing $f(z)$ as $$\frac{1}{(1-(x+iy))^2}$$ and then I expand the bottom to get $$\frac{1}{(1-2x+x^2-y^2) + i(2y-2xy)}$$ The answer says ...
5
votes
1answer
78 views

Intuition behind $i^{i}$.

My query is about the $i^{i}$ , where $i$ is defined to be the imaginary unit, and $i \in C$. I know the proof of this value, we just have to substitute $i$ as ...
1
vote
0answers
28 views

Finding the RHS of a complex equation

Consder $z$, a complex number, such that $z+\frac{1}{z} = 2\sin(a)$, $a \in (0,2 \pi)$ . Find : $$ z^{4n} + \frac{1}{z^{4n}} = ? $$ I tried expanding it in trig form, then by applying de Moivre's ...
2
votes
1answer
27 views

How does uncertainty propagate through an equation with complex variables?

I am trying to understand how uncertainty propagates through systems with complex variables. Given the general error propagation formula $$ \sigma^2_u = \left(\frac{\partial u}{\partial ...
7
votes
2answers
91 views

How to visualize $\mathbb{C}^2$?

In a homework question I had to do, the rotational matrix $A = \begin{pmatrix} 0&-1\\1&0 \end{pmatrix}$ was given. Its eigenvalues in $\mathbb{C}$ are $i$ and $-i$. The set of all eigenvectors ...
-4
votes
1answer
22 views

Find the priniciple argument of $-5\sqrt{2}+5i\sqrt{2}$ [on hold]

Find the priniciple argument of $-5\sqrt{2}+5i\sqrt{2}$ must be the exact value and in the form $z=...\pi$
0
votes
0answers
9 views

Is there a particular order when evaluating the complex exponential?

This question might seem silly. But I just cannot get this: $e^{i 6\pi} = 1+0i$ ($e^{i 6\pi})^{1/2} = \sqrt{1+0i} = \sqrt{1} = 1$ ($e^{i 6\pi})^{1/2} = e^{i3\pi} = -1+0i = -1$
-1
votes
1answer
41 views

If $\sin{x}+\sin{y}+\sin{z}= \cos{x}+\cos{y}+\cos{z}=0$, find the value of $\cos{2x}+\cos{2y}+\cos{2z}$. [on hold]

Is there any way to solve this question using complex numbers? I am trying the general way too but I am unable to solve the question.
1
vote
5answers
47 views

Complex Numbers Roots of Unity

By multiplying two roots, one is the conjugate of the other, we get one. Does someone know why and proof that. Many thanks
16
votes
5answers
1k views

Proving the following number is real

Let $z_i$ be complex numbers such that $|z_i| = 1$ . Prove that : $$ z\, :=\, \frac{z_1+z_2+z_3 +z_1z_2+z_2z_3+z_1z_3}{1+z_1z_2z_3} \in \mathbb{R} $$ This problem was featured on my son's final ...