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

learn more… | top users | synonyms

1
vote
2answers
8 views

Electrical Engineering (complex numbers)

Electrical Engineering ($j=i=\sqrt{-1}$): $$H_v(\omega)=\frac{R}{R+\frac{1}{j\omega C}}=\frac{j\omega CR}{1+Rj\omega C}$$ And we know that: $\omega_0=\frac{1}{RC}\Longleftrightarrow ...
0
votes
3answers
17 views

Complex plane (Show that triangle is right-angled)

The points $O$,$P$ and $Q$ in the complex plane represent the complex numbers $0+0i$, $4+2i$ and $3-i$ respectively. Find the exact length of $PQ$ and hence, or otherwise, show that triangle $OPQ$ is ...
1
vote
0answers
23 views

Complex Number (Angle)

The complex number $z$ is given by $z=-2+2i$ Find the modulus and argument of $z$ Write down the modulus and argument of $\frac{1}{z}$ Show on an Argand diagram the points A,B and C representing the ...
0
votes
1answer
27 views

Complex Numbers (Find p and q)

The complex numbers z1 and z2 are given by $$z_1=5+i,z_2=2-3i$$ Determine the values of the real constants $p$ and $q$ such that $$\frac{p+iq+3z_1}{p-iq+3z_2}=2i$$ My attempt, I substitute $z_1$ and ...
0
votes
3answers
107 views

Complex Numbers (Modulus)

The complex numbers $z_1$ and $z_2$ are given by $$z_1=5+i,z_2=2-3i$$ Find the modulus of $z_1-z_2$ My attempt, modulus of $z_1-z_2=\sqrt{5^2+1^2}-\sqrt{2^2+3^2}$ $=\sqrt{26}-\sqrt{13}$ ...
0
votes
1answer
19 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
27 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: ...
-1
votes
3answers
38 views

Cube roots of complex numbers [on hold]

I need help with finding the cube roots of the complex number 27... I know that the obvious answer is three, but what is the less simple method to solving this?
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
19 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
17 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
27 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
21 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
175 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
26 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
82 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
28 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
130 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
79 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 ...