0
votes
3answers
42 views

Algebraic Equation?

$$Ve^{i\theta} = We^{i\phi}$$ where, $V$ and $W$ are some real constants. From this my book concludes: $\theta = \phi$. How does it conclude this? I don't see why its valid to just equate the ...
0
votes
1answer
53 views

Complex number, strangely written

Find all the complex solutions of the equation: $$\frac{z^3}{i} = 1$$ I mean is this the same thing as $$z^3 = i$$? Because I don't understand why my teacher would put it like that on a test. At ...
0
votes
2answers
30 views

Finding numbers $a$ and $b$ for a complex number

Problem. Given a complex number $$z=2-2i$$ Find numbers $a$ and $b$ such that $$a+ib = \frac{1}{z}$$ I tried multiplying both sides by $z$ and got $$(a+ib)(2-2i)$$ $$= 2a-2ai+2bi-2bi^2$$ ...
0
votes
1answer
69 views

Why isn't $i$ affected by powers?

When finding roots of complex functions we can write for example: $$z=2-2i$$ Let's find complex numbers $w$ such that $$w^4 = 2-2i$$ $$\large z = \sqrt{8} e^{ \frac{- \pi }{4} i}$$ This reads: ...
0
votes
2answers
31 views

Finding complex roots

Problem Find the roots of $$z^3 = -1 - i$$ And calculate $$ \sqrt[3]{-1-i}$$ I'm looking at the solution outlined in my book but I'm having problems understanding it. I can find the length and ...
4
votes
1answer
31 views

Unsure with second order complex differential equations

Solve $$y'' - 4y' + 5y = 0 $$ Where $y(0) = 0 \ , \ y'(0) = 2$. So I solve this as a second degree polynomial (no idea why) $$\frac{4 \pm \sqrt{16-20}}{2} = 2 \pm 2i$$ So the CASE III solution as ...
0
votes
2answers
37 views

Using the formulas de Moivre to deduce trigonometric identities.

Yesterday I made a test of complex variables, and this contained a question (in which I could not solve) that asked to use the de Moivre formulas to deduce the following trigonometric identities: ...
0
votes
1answer
24 views

Product of 2 couples of complex numbers

Let $a, b, c, d$ be complex numbers, but such that $b = \displaystyle \frac{a}{k}, d = \displaystyle \frac{c}{k}$ with $k$ real. Moreover, $$ab^* = \frac{|a|^2}{k} = cd^* = \frac{|c|^2}{k} = r$$ ...
0
votes
1answer
52 views

proving a limit of a function by definition

Consider $f: \Bbb{C} \to \Bbb{C}$ defined by $$ f(z) = \begin{cases} z^3 + 2z &\text{if } z \ne i \\ 3 + 2i &\text{if } z = i \end{cases} $$ Prove that $$ \lim_{z \to i} f(z) = i $$ using the ...
1
vote
1answer
26 views

Integral calculus use of Newton-Leibnitz rule

My friend asked me this question: If $y(x)= \int_{0}^{x}f(t)\sin{(px-pt)}dt$ then what is the value of $y''(x)-((p^2)*y(x))$. He gave me the hint to consider $\sin(px-pt)$ as the imaginary part of ...
0
votes
0answers
22 views

Providing an upper bound for a sum of complex numbers

Let $(\alpha_l)_{l=0}^{k-1}$ and $(\beta_l)_{l=0}^{m-1}$ be two sequences of complex numbers where $m>k$. It is known that $$ 0<\frac{1}{m}\sum_{l=0}^{m-1}|\beta_l|^2\leq A $$ where $A>0$. ...
-1
votes
1answer
23 views

Solve equation a complex equation completely

I need some help to solve the following equation. Equation $z^4 - 4z^3 + \frac{5}{2}z^2 - z + \frac{1}{4} = 0$ has an solution $z = \frac{1}{2}i$. Solve the equation completely.
3
votes
2answers
64 views

Justifying an ODE's solution

In an introductory lesson into ODEs, in order to "semi-rigorously" justify the solution for e.g. : $(a)\ \ y'+y=0$ we proceed without an ansatz or guess solution (hence the "semi-rigour"): Let: ...
2
votes
2answers
45 views

Solve the equations $z^2 + (2 - 2i)z + 2i = 0 $ by completing the square

I tried solving this thing by completing the square and I always end up with something like this $(z^2 + (2 - 2i)z - 2i) + 2i + 2i = 0 $ and it doesn't seem like to me that you can factor the part in ...
2
votes
1answer
33 views

Behaviour of generating function for distinct partitions

We all know that the generating function for distinct partitions is $$Q(x)=\prod\limits_{k=1}^{\infty}(1+x^k)$$ Computation on Maple suggests that $\lim\limits_{x\to1^-}Q(-x)=0$, $\lim\limits_{x\to ...
3
votes
1answer
127 views

Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this: The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The ...
2
votes
1answer
31 views

How is this computed?

$ \left| 1-e^{-is\lambda} \right|^2 = 2 (1-cos\lambda s)$ where $i=\sqrt{-1}$ I don't know how to work with $i$. Thanks
5
votes
3answers
152 views

$n$th derivative of $e^x \sin x$

Can someone check this for me, please? The exercise is just to find a expression to the nth derivative of $f(x) = e^x \cdot \sin x$. I have done the following: Write $\sin x = \dfrac{e^{ix} - ...
3
votes
2answers
96 views

An apparently elementary fact about complex numbers.

Let $f,g\in \mathbb{C}$. My textbook states that it is an elementary fact that $$\lim_{t\to 0,t\in\mathbb{R}}\frac{|f+tg|^p-|f|^p}{t}=\frac{p}{2}|f|^{p-2}(\overline{f}g+f\overline{g})$$ I don't know ...
1
vote
0answers
59 views

Calculating Fourier magnitude spectrum for Local Binary Pattern histogram

I have the follwoing discrete Fourier transform function defined in my book (Computer Vision using Local Binary Patterns, Pietikainen et. al, 2011): $$H(n, u ) = \sum_{r=0}^{P-1}c_{nr} ...
1
vote
5answers
121 views

Finding real and imaginary part of exponential function

Can someone explain to me how I find the real and the imaginary part of $e^{\theta i}$? I'm learning complex numbers but I don't quite understand how $e$ is intertwined in all this.
1
vote
1answer
59 views

Complex root won't work

So I'm trying to get this: http://www.wolframalpha.com/input/?i=%288*sqrt%283%29%29%2F%28z%5E4%2B8%29%3Di And I've calculated $z^4=16 \left( \cos (\frac{- \pi}{3})+ \sin ( \frac{- \pi}{3}) \right)$ ...
-1
votes
2answers
36 views

Complex roots problem [duplicate]

I've got a complex equation with 4 roots that I am solving. In my calculations it seems like I am going through hell and back to find these roots (and I'm not even sure I am doing it right) but if I ...
2
votes
1answer
53 views

How come complex numbers represent coordinates?

I'm wondering why complex numbers represent coordinates without being on the form of a tuple (a,b). The complex numbers come in the form: $a+bi$ where $a$ denotes the real part and $bi$ denotes the ...
0
votes
1answer
34 views

Finding argument in complex numbers

I am having troubles with finding the argument in complex numbers(except for the obvious ones). Is there an easy or algorithmic way to find the argument? Does anyone have a mnemonic to find them?
1
vote
1answer
54 views

Calculus $T_1=\prod_{k=1}^{n-1} \cos\frac{k\pi}{2n}$

Calculus: $$T_1=\prod_{k=1}^{n-1} \cos\frac{k\pi}{2n}$$ and $$T_2=\prod_{k=1}^{n-1}\sin\frac{k\pi}{2n}$$ My tried: I use Euler's formal: $$z_k=e^{i\frac{k\pi}{2n}}=\cos\frac{k\pi}{2n}+i\sin ...
0
votes
2answers
85 views

How do I solve this integral?

As stated the title, I get to a point which I can't do anything, and I'm sure I've made a mistake some where, here is my full working out: $$ \int e^{ix}\cos(x)dx \\ u = e^{ix} \text{ | } u'= ie^{ie} ...
6
votes
2answers
150 views

Why isn't $\int\sin(ix)~dx$ equal to $i\cos(ix)+C$ ?

I was playing around with imaginary numbers, and I tried to solve $$\int\sin(ix)~dx$$ and ended up getting $$i\cos(ix)+C$$ But apparently the answer is $$i\cosh(x)+C$$ I was just wondering, is this ...
2
votes
3answers
133 views

Limit n tends to infinity

How can i solve this: $$ \lim_{n\to\infty} \cos(1)\cos(0.5)\cos(0.25)\ldots \cos(1/2^n) $$ I tried using comlex numbers and logarithms but did'nt work out.Can anyone help please.
0
votes
0answers
51 views

Solving physics problems using real and imaginary numbers

I was working on a particular physics problem and like we usually in physics do - replaced $\cos(x)$ with $e^{ix}$ and worked the result. When I tried to solve it without complex numbers I stuck. In ...
4
votes
1answer
32 views

complex integral problem

I have to evaluate this $$\int_{c} \dfrac {|z| e^{z} }{z^2}$$ where C is the circunference with radius 2. I have tried to apply the Cauchy formula but $|z|e^{z} $ is not holomorfic. I know the resut ...
2
votes
1answer
246 views

Proof that $\frac{1}{2}(c_{n}-d_{n})\pi=1$ if $n$ is odd, for $f(z)=\csc(z)$ and $\{c_{n}\}$ and $\{d_{n}\}$ Laurent coefficients of $f$

Let $f(z)=\csc(z)$ and $\{c_{n}\}$ and $\{d_{n}\}$ Laurent coefficients of $f$ in $\{z\in \mathbb{C}:0<|z|<1\}$ and $\{z\in \mathbb{C}:1<|z|<2\}$ respectively. Proof that ...
1
vote
1answer
84 views

How to prove the following inequality of logarithm?

Let $x,y,z\in\mathbb{C}.$ Suppose $$z=\frac{1}{2}(xy\pm\sqrt{x^2y^2-4(x^2+y^2)} ).$$ Show that $$log^+|z|\leq log^+|x|+log^+|y|+log 2.$$ Where $log^+\phi=max\{0,log\phi\}.$ Here we are also ...
13
votes
1answer
77 views

Simplification of a trilogarithm of a complex argument

Is it possible to simplify the following expression? $$\large\Im\,\operatorname{Li}_3\left(-e^{\xi\,\left(\sqrt3-\sqrt{-1}\right)-\frac{\pi^2}{12\,\xi}\left(\sqrt3+\sqrt{-1}\right)}\right)$$ where ...
0
votes
1answer
34 views

$-ia(1\pm \sqrt{1-1/a^2})$, $a>0$ inside unit circle?

Given $a>0$ I would like to know whether: $\alpha=-ia(1+ \sqrt{1-1/a^2})$ and $\beta =-ia(1- \sqrt{1-1/a^2})$ are inside the unit circle. How can I check that?
0
votes
2answers
139 views

Rewriting $x^3-3xy^2+2xy+i(-y^3+3x^2y-x^2+y^2 )$ in terms of $z$, with $z=x+yi$

How do I write $f=u+iv$ with: $u=x^3-3xy^2+2xy$ and $v=-y^3+3x^2y-x^2+y^2 $ in terms of $z$ with $z=x+yi$?
0
votes
3answers
114 views

Evaluate $\oint_C\frac{dz}{z-2}$ around the square with vertices $3 \pm 3i, -3 \pm 3i$.

I'm having a tough time figuring out $\gamma(t)$ in $\oint_C f(z) dz = \int_a^b f(\gamma(t)) \gamma '(t) dt$. I think I have to split this into four parts, the four separate lines, but from there I'm ...
0
votes
1answer
95 views

Evaluate $\oint_C |z|^2 dz$ around the square with vertices at $(0,0), (1,0), (1,1), (0,1)$

I don't think I quite understand how to go about this. My solution so far: $\oint_C |z|^2 dz = \oint_C (x^2 + y^2)dz = \oint_C (x^2 + y^2) d(x+iy) = \oint_C x^2 + y^2 dx + i\oint_Cx^2+y^2dy$. ...
2
votes
1answer
116 views

when point lies outside what does cauchy integral formula state ??

I have the following complex line integral: $$ \int_{|z| = 2} \frac{z}{z - 3} $$ My prof said it is 0,but did not explain.He just said that the point 3+0*i lies outside the circle. But the cuachy ...
2
votes
2answers
212 views

Integrate $\int_{0}^{2\pi} \frac{R^{2}-r^{2}}{R^{2}-2Rr\cos \theta +r^{2}} d\theta= 2\pi$, by deformation theorem

I need prove that: $$\int_{0}^{2\pi} \frac{R^{2}-r^{2}}{R^{2}-2Rr\cos \theta +r^{2}} d\theta= 2\pi$$ By deformation theorem, with $0<r<R$. Professor gave us the hint to use the function ...
2
votes
2answers
223 views

Calculate the integral $\int_{0}^{2\pi}\frac{1}{a^{2}\cos^2t+b^{2}\sin^{2}t}dt$, by deformation theorem.

I want to prove: $$\int_{0}^{2\pi}\frac{1}{a^{2}\cos^2t+b^{2}\sin^{2}t}dt=\frac{2\pi}{ab}$$ by the deformation theorem of complex variable. Then I consider a parameterization ...
2
votes
2answers
59 views

write $\lvert 10+4i\rvert$ in the form $z=re^{i\theta}$

write $\lvert 10+4i\rvert$ in the form $z=re^{i\theta}$ I am not sure what to do with the absolute value in this case.
1
vote
1answer
49 views

How to solve $ \begin{cases} \cos (z_1 +iz_2) = i\\ |z_1|=|z_2| \end{cases} $?

How to solve $ \begin{cases} \cos (z_1 +iz_2) = i\\ |z_1|=|z_2| \end{cases} $? where $z_1, z_2$ are complex variables Rectangular form is convenient for the first equation, and polar form is ...
5
votes
1answer
113 views

Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ .

Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ ? How to find if it is convergent or not? Thanks!
12
votes
2answers
315 views

Does $z^i=i^z$ have any solutions, beside $z=i$?

Does this equation have any solutions: $$\large{z^i=i^z}$$ Putting polar form of $z$ is better for LHS, But rectangular form is suitable for RHS ! What to do? Thanks!
25
votes
5answers
1k views
2
votes
1answer
93 views

Uniquely defined function

I am struggling with the following exercise: Let $f:B(0,1) \rightarrow \mathbb{C}$ be a holomorphic function and we have $\forall n \in \mathbb{N}_{\ge 2}: f'(\frac{1}{n})=f(\frac{1}{n})$ then f can ...
7
votes
1answer
92 views

Why does $\sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )+{1\over2}-{1\over3} = \gamma$?

How could one prove that $$x = \sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )$$ is such that $x+{1\over2}-{1\over3} = \gamma$ ? I am having problems just calculating ...
0
votes
1answer
221 views

making the domain of $z ↦\tan(z)$ injective

Given the following: $\sin(z)$ = ($e^i$$^z$ - $e^-$$^i$$^z$)/$2i$ $\cos(z)$ = ($e^i$$^z$ + $e^-$$^i$$^z$)/$2$ $\sin(z)\cos(w) - \cos(z)\sin(w) = \sin(z-w)$ $\sin(z) = 0$ has solution $z = kπ$ for ...
1
vote
1answer
124 views

Where is there a good introduction to hypercomplex numbers and calculus?

I'm looking for a good introduction to hypercomplex numbers that requires as little math knowledge as possible, yet covers hypercomplex numbers as thoroughly as possible. I'm interested specifically ...