2
votes
2answers
56 views

Different Definitions Of The Sine Function

I was hoping someone could give me a flow chart or high-level map connecting all of the definitions of the sine function, with some of the reasons why we care next to each. I've tried this but I'm not ...
1
vote
1answer
50 views

Rewriting a double integral with complex exponential function

Why can we write $$ \begin{align} I_T &= \int_\mathbb{R}\int_{-T}^{T}\frac{e^{-ita}-e^{-itb}}{it}e^{itx}dtdF(x)\\ &= \int_\mathbb{R}\left[\int_{-T}^{T}\frac{\sin(t(x-a))}{t}dt - ...
1
vote
2answers
28 views

evaluate exponential using Euler identity

let us consider following exponential $e^{-j*\pi*k/2}$ and $e^{j*\pi*k/2}$ we can decompose it as $cos(\pi*k/2)-j*sin(\pi*k/2)$ and second one same with plus sign ...
1
vote
4answers
60 views

Need an example of piece wise function continuous but not differentiable

I Need an example of piece wise function continuous but not differentiable. One of the functions has to be trigonometric and the other has to be exponential. Please
2
votes
4answers
59 views

Can somebody explain to me why these terms are equal?

I just read a proof on ProofWiki that proves Euler's formula, but I can't seem to understand what is done in this following step: ...
0
votes
2answers
61 views

No. of real solutions of the equation $2 \cos (\frac{x^2 + x}{6}) = 2^x + 2^{-x} $

How many real solutions are there of the equation $2 \cos (\frac{x^2 + x}{6}) = 2^x + 2^{-x} $? Please illustrate.
1
vote
2answers
104 views

Fourier Transform the following exponential and cosine function: $f(x) = e^{-a^{2}x^{2}}cos(bx)$

I have a previous exam here for my course (Provided by the professor) that requires me to do a Fourier Transform of the following equation. Here is the function: $f(x) = e^{-a^{2}x^{2}}cos{(bx)}$ ...
0
votes
2answers
44 views

Expressing the sine function in terms of exponential

Prove $e^{iz} - e^{-iz} = \sin z$. I used $$\begin{align*} \sin z & = z - z^3/3! + z^5/5! - z^7/7! + \dots & (i) \\ e^{iz} & = 1 - z^2/ 2! - iz^3/3! + \dots & (ii) \\ e^{-iz} ...
2
votes
1answer
35 views

How to simplify $\cot(\sec^{-1}(e^x))$

I've been trying to simplify $\cot(\sec^{-1}(e^x))$. I thought substitution might the way to go about it so I said: let $u = \sec^{-1}({e^x})$ I'm therefore trying to find $\cot(u)$ From $u = ...
0
votes
1answer
38 views

Simplify this expression?

I have the following expression $$\frac 12 x_0e^{-\beta t}\left[\left(\frac {\beta}{i \sqrt{\omega ^2-\beta ^2}}+1\right)e^{i \sqrt{\omega ^2 - \beta ^2}t}+\left(\frac {- \beta}{i \sqrt{\omega ^2 - ...
9
votes
2answers
195 views

Conjectural closed form for $\int_0^\infty\sqrt[3]z\ \operatorname{Ei}^2(-z)\,dz$

While trying to answer the question "A closed form for $\displaystyle\int_0^1\frac{\ln(-\ln x)\ \operatorname{li}^2x}{x}dx$", I came up with a conjecture: $$\int_0^\infty\sqrt[3]z\ ...
1
vote
2answers
213 views

Rewriting exponential function using Euler

I have the following function that I want to express using trigonometric functions: $$f(x) = \frac{1}{2}\left(\frac{2i+2}{2i+1}e^{ix} + \frac{-2i+2}{-2i+1}e^{-ix}\right)$$ I have come so far as: ...
1
vote
5answers
142 views

How does $Ae^{4ix}+Be^{-4ix}=A\cos(4x)+B\sin(4x)$?

$e^{ix}=\cos(x)+i\sin(x)$ $Ae^{4ix}=A(\cos(4x)+i\sin(4x))$ $Be^{-4ix}=B(-\cos(4x)-i\sin(4x))$ What am I doing wrong? I am trying to find the complimentary function of $\frac{d^2y}{dx^2} ...
0
votes
1answer
107 views

Location of roots of certain transcendental equations.

If $a$ and $b$ are two solutions of $\,e^x \cos x -1=0$, then how many solutions of the equation $ e^x \sin x-1=0$ lie between $a$ and $b$ ?
5
votes
1answer
120 views

Interesting definite integral involving exp and trig

I'm trying to evaluate the following integrals: $$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \cos(\phi) d\phi$$ $$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \sin(\phi) d\phi$$ for which I want to find ...
2
votes
1answer
87 views

Why is $e^{g(x)} = \pi$ where $g(x)$ is holomorphic in Weierstrass factorization of sine function?

Why is $e^{g(x)} = \pi$ where $g(x)$ is holomorphic in Weierstrass factorization of sine function? I just can't get why it's true.
2
votes
3answers
154 views

Incoherence using Euler's formula

Using the relation $\ e^{ix} = \cos(x) + i\sin(x)$ and substituting for $\ x = \pi$, we have the well-known Euler identity, $ e^{i\pi} = -1$. Substitute also for $ x = -\pi $, we have $ e^{-i\pi} = ...
1
vote
2answers
408 views

Evaluating a double integral involving exponential of trigonometric functions

I am having trouble evaluating the following double integral: $$\int\limits_0^\pi\int\limits_0^{2\pi}\exp\left[a\sin\theta\cos\psi+b\sin\theta\sin\psi+c\cos\theta\right]\sin\theta d\theta\, d\psi$$ ...
6
votes
1answer
289 views

integral of the product of a trigonometric and an exponential function

Since tan has an odd power I would normally aim to sub $u=\sec(x)$, but I cant get rid of the $2^x$. $$\int 2^x \tan^9(x^2)\sec(x^2)dx$$ I also tried integrating by parts but it got more complicated. ...
0
votes
0answers
80 views

How would I go about solving this Euler's Equation problem, getting even and odd components?

I'm stuck on this question in my signals and systems class, the question asks to find the even and odd components of the equation. Now I know that $e^{jx} = \cos(x) + j\sin(x)$, however this ...
0
votes
1answer
46 views

How to solve the following special inequality?

Find $k$, as a function of $d_2$ and $d_3$, such that: $$\left \vert { d_2 \left [ \sin(e^{d_3\,y}) - \sin(e^{d_3\,x})\right] + (x-y) d_2 d_3 e^{d_3\,z} \cos(e^{d_3\,z})} \right \vert \le k ...
5
votes
1answer
147 views

How to evaluate $\frac{1}{b^2}\int_0^\infty z^{-2}\exp(-a z)\sin^2(b z)\, \mathrm dz$?

How can I integrate the following: $$\frac{1}{b^2}\int_0^\infty z^{-2}\exp(-a z)\sin^2(b z)\, \mathrm dz$$ for $a,b>0$? Maple gives a compact result: $$\frac{1}{b} \tan^{-1}(c) - \frac{1}{ac^2} ...