Fourier transform involving a delta function For ease let us suppose that I am trying to transform the function $f(t)=\delta(a^2 -t^2)$. Therefore the Fourier transform is given by $$\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \delta(a^2 -t^2)\exp(-i\omega t)\,dt.$$  My question is how to evaluate $\int_{-\infty}^{\infty}\exp(-i\omega t)\,dt$, and how it relates to the delta function? I know that the Fourier transform of $\delta(t)$ is unity, but how does this change when considering  $ \delta(a^2 -t^2)$ ?  Thanks very much.
Calculating the Fourier transform of $f(t)=Ae^{-i\omega_0 t}$ I am trying to calculate the Fourier transform of $f(t)=Ae^{-i\omega_0 t}$  I'm getting an infinity which is giving me problems. Here are my steps:  $$F(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(t)e^{i\omega t}dt$$ $$=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty Ae^{-i\omega_0 t}e^{i\omega t}dt$$  $$=\frac{A}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{t(i\omega - i\omega_0)}dt$$ $$=\frac{A}{\sqrt{2\pi}(i\omega - i\omega_0)} |e^{t(i\omega - i\omega_0)}|_{-\infty}^{\infty}$$ $$=\frac{A}{\sqrt{2\pi}(i\omega - i\omega_0)}(\infty - 0)$$  Where am I going wrong? Thanks.
How can I lock a file using java (if possible) Can an Indian travel to India with 50 days before passport expires? My wife is travelling from Philadelphia to India in January 2019 and, on her outbound trip, she will only have 50 days validity left on her passport. However, while in India, she will be renewing her passport. The reply from a similar question seems to answer point 1 below, but I wanted a second opinion to be sure.  Here are the questions we have:  Can she travel from the US to India via Doha, Qatar with passport validity of only 50 days? Can we make a reservation on Qatar Airways for the roundtrip travel using her current passport and before the return from India to the US update passport information on the reservation? $\lim\limits_{n \to \infty} \frac{a_n}{b_n}=l$ ; Prove $\sum\limits_{k=1}^\infty a_k$ converges iff $\sum\limits_{k=1}^\infty b_k$ converges. Written, in Ex.8 Ch.9.1 of the book Advanced Calculus by P. M. Fitzpatrick :     Suppose that $\sum\limits_{k=1}^\infty a_k$ and $\sum\limits_{k=1}^\infty b_k$ are series of positive numbers such that $$\lim_{n \to \infty} \frac{a_n}{b_n}=l \ \ \ \text{and} \ l&gt;0.$$ Prove that the series $\sum\limits_{k=1}^\infty a_k$ converges iff the series $\sum\limits_{k=1}^\infty b_k$ converges.   Am I correct by the following (sketch of) proof? :  1- For a given $\epsilon_1$ there is $N_1$ such that $\left|\frac{a_n}{b_n} - l\right| &lt; \epsilon_1$ for all $n \ge N_1$.  2- Since the series $\sum\limits_{k=1}^\infty a_k$ converges, for a given $\epsilon_2$ there is $N_2$ such that $\left|b_{n+1}+\dots+b_{n+k}\right|&lt; \epsilon_1$ for all $n \ge N_2$ any for all natural numbers $k$.  3- Define $N = \max {\{N_1,N_2}\}$.   4- From Step (1), $a_{n+k} &lt; (\epsilon_1+l) b_{n+k}$ for all $n \ge N$ any for all natural numbers $k$. [Also, $a_i$'s and $b_i$'s are all positive]. Thus $a_{n+1}+\dots+a_{n+k} &lt; (\epsilon_1+l) (b_{n+1}+\dots+b_{n+k})&lt; \epsilon_3$. Then the convergence of the series $\sum\limits_{k=1}^\infty b_k$ implies the convergence of the series $\sum\limits_{k=1}^\infty a_k$.  5- For the reverse implication, we use the fact that $\lim\limits_{n \to \infty}\frac{a_n}{b_n}=l \iff \lim\limits_{n \to \infty}\frac{b_n}{a_n}= \frac{1}{l}= l' &gt;0$ and repeat the process this time for a and b exchanged.   6- The Quotient Property For Sequences hold for a nonzero limit in the denominator, but since the limit in the numerator also is zero so we may use the The Quotient Property For Sequences.    Thanks. I just replaced a broken light fixture in my closet. (The switch in the light fixture broke when I pulled too hard on the chain.)   When replacing the light fixture I noticed that there is only 30 V AC between the white and black wires. The ground wire is cut back and is not being used. By the way, the wiring in my house is all aluminum.   I know the new light fixture works. I checked it out with my multimeter, and I tried it in a different closet. But the light bulb won't turn on because there is only 30 V. (Even if I disconnect the light fixture completely there is still only 30 V.   It seems to me that it must be a problem with the wiring.  My attic does provide some (difficult) access to the wiring, but I am not sure where to start looking for the problem or what to look for. Can anyone offer some suggestions? Why does Python's itertools.cycle need to create a copy of the iterable? Name of mean absolute error analogue to Brier score? Yesterday's question  got me curious about probability scoring. The  $$\frac{1}{N}\sum\limits _{i=1}^{N}(\text{prediction}_i - \text{reference}_i)^2$$ is a mean squared error measure. Does the analogous mean absolute error performance measure $$\frac{1}{N}\sum\limits _{i=1}^{N}|\text{prediction}_i - \text{reference}_i|$$ have a name, too? Do spells with a range of "Self" target the caster? New to D&amp;D, and thought about creating a Beast Master. I was wondering if spells with a range of "Self" would count you as the "target" of the spell. There's some debate in my group, as unlike something such as Cure Wounds, which specifies "A creature you touch," there's technically no mention of a target in a Self range spell within the description. A fair amount of people in my playgroup come from a TCG background, where the presence (or lack thereof) of keywords like "target" are essential for determining interaction.  tl;dr Would Paladin Smites or Thunderwave count as "targeting" yourself, and therefore be affected by Beast Master's Share Spell? Show that any bounded linear functional on a normed linear space is continuous Show that any bounded linear functional on a normed linear space is continuous.   Can we say that it is uniformly continuous ?   Also, is it true, if we reverse the statement    any continuous  linear functional on a normed linear space is bounded ?? What is the difference between "citizen" and "denizen" Citizen: 1. A legally recognized subject ornational of a state or commonwealth, either native or naturalized. 2. An inhabitant of a particular town or city.  Denizen: 1. An inhabitant or occupant of a particular place.  Same thing? I noticed that when I had a helium filled, latex balloon inside of my car, it moved forward in the cabin as I accelerated forward. The faster I accelerated forward, the faster the balloon went from the back of the car to the front of the car. The balloon didn't have a string.  This became a game with my 4 year old as we drove home. We figured out where the balloon would go based on how fast I accelerated, turned corners etc.  I expected that it would act a lot like the water in a cup does, but it was the total opposite it seemed.  What forces caused this behavior?  I assumed it has something to do with the fluid dynamics in the closed cabin, but I can't figure it out. Console flashes on launching, then closes (ms-windows) I installed blender on a windows 7 64 bit system.  I opened it one time then went to the tutorials.  in the meantime I had a power failure and when I tried to start blender again I get a dos window that closes itself then nothing.  I reinstalled Blender and get the same thing.  I am going to try to uninstall then reinstall but in the mean time, any other suggestions are welcome. Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$ Remove the shortcut of an application that was installed using Wine from the Unity Dash How do I remove an application shortcut from the Unity Dash (not from the Unity Launcher)?  My specific issue is I tried to install a game using Wine a while ago. I then uninstalled the game, then uninstalled Wine. Today I found that shortcuts to the game (and other related content that came with it) remain when I push Super and type the game name. (Wine is not visible though.) How to compare two TeX codes and mark differences with color I would like to find easy tool, which compares two TeX codes and generates a PDF file of the first code, in which the differences from the second code are marked by a given color. I use winedt 7.0 under windows 7.  I just noticed that latexdiff provides what I need. As far as I understand it is included in Miktex. Is there an interface (menu commands) in winedt 7.0, which allow to work with latexdiff ? Orthonormal basis with specific norm I need to show that $v_1,...,v_n$ is basis for $V$ whenever $e_1,...,e_n$ is an orthonormal basis for V and $v_1,...,v_n$ are vectors in $V$ such that $$\left\Vert e_i-v_i\right\Vert &lt; \frac{1}{\sqrt{n}}.$$ I know already that I only need to prove that they are lineary independent. Also I tried to prove it by contradiction, but I got stuck fairly fast. Any help, or hint how to prove it (with ot without contradiction) will be appriciated. How to measure supply currents ranging from 1μA to 10mA? How to reconcile the cashless human/Federation society with human traders and merchants Effective Refractive Index in Paper I am reading the paper, entitled "Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings" by Eric B. Grann, M. G. Moharam, and Drew A. Pommet.   In the paper they mention this formula:        Do you know how it can be derived and where it comes from ? When is logistic regression solved in closed form? Best way to find all factors of a given number