Analysis and Calculus

I seek a proof that xn -1 = 0 has a primitive root.

Exactly what the title says. Not an easy function to work with, for some reason it doesn't have a known indefinite integral even though it seems like it almost should. The answer seems to have something to do with the polygamma function though the exact process of differentiating the gamma function seems non-existent on the internet.

So as some here may know, I've been teaching myself calculus. I learned differentiation and finding tangent lines to curves using infinitesimals with a little help from my friend's calc book; then about a year later I finally grasped the fundamental theorem of calculus after someone pointed me to an illustration on wikipedia which showed that the addition of an infinitely thin rectangle in relation to a section of area under a curve ends up working out to differentiation, except giving you the function for the curve instead of a derivative, meaning that inverse differentiation is used for finding the area under a curve. Then after that, with much struggle and confusion t…

For arccos(x), outside of its domain on [-1,1], it turns into i*arccosh(x). But, if you took the inverse of accros(x) (no h), those parabola-like structures caused by i*cosh(x) disappear when that function is converted into the function cos(x). I want to know why that I*cosh(x) component suddenly disappears when I flip arccos(x) over the x axis to find the inverse, or if they are still there but just not being displayed?

It is an integral factorial of 7 or 6 or 5 (6!). A description for connectivity of the main Sigma used ...... A description for Interchange ... A description for terminology .... A description for multiplicativity .... network grid ......... Salvation ........ Tree ........... and Division! Compensation Validity dictates that Solution Nature should be used in this study. Just an Introduce: Where is The Main Divisions of Mathematics ........... Just not to support The EVIL SIDE!!!!!!! Administration. Example: Volume Integrals of (1/r) doesn't 'converge' to one solution. Let's begin and follow: Find: SSS gaussian-order-distribution/(R_p) dX_s dY_s dZ_s

I'm running into a problem where I want to represent a series of radians as A*arccos(Bx+C)+D, like for instance pi/2, 3*pi/4, pi...18*pi/12.. or pi/2, pi, 3pi/2, 2pi, pi/2, pi, 3pi/2....ect. But, because the domain of arccos is limited and then becomes some imaginary hyperbolic trig function outside of the domain 0<x<pi, I'm having trouble figuring out how to do that. Like normally, you could represent a series of cyclical numbers like 1,-1,1,-1 as cos(pi*x). I want to do the same thing with radians using arccos(x) where it just keeps repeating periodically as x grows arbitrarily large

Overall there doesn't seem to be a lot on multifactorials. Generalized formulas for multifactorials seem to only work for numbers evenly divisible by k, where k is the type of factorial "like double, triple, quadruple ect). But, my concearn is that the non-evenly-divided numbers are treated completely arbitrarily and not even conjectured, just arbitrary selected. For instance, 7!! (double factorial) would count down to 7*5*3*1, whereas 8!! would count as 8*6*4*2. See the difference? 8!! ends on 2, like its suppose to, and its evenly divisible by 2. 7!! however ends on 1, like its not suppose to because its not a mono-factorial. But, is it really "not suppose to" or is the…

I've stumbled upon a direct proof on Reddit, which uses differentiation to solve an equation. See https://medium.com/criminal-clouds/hydrostatic-lapse-part-1-of-3-e8a1534cd12d At particular point we reach this equation: This equation is then "logged" on both sides: then differentiated on both sides: then rearranged: then differentiated on both sides again and solved: Is this the correct way to solve this equation? I'm no maths expert, but this differentiation of both sides seems like far too much of a short cut.

What is the limit of (1-1/n)^n? For (1+1/n)^n it is e, about 2.7 but I can't find the negative result.

The standard proof that a polynomial cannot be irreducible if it has repeated roots uses calculus (differentation). I would like to find a proof without calculus . . .

I have seen asserted that for polynomials f and h, f divides h implies that fbar divides hbar, where "bar" indicates reduction modulo p, a prime number say. But when I test this with sample polynomials, it does not seem to be true. For example modulo 3 and f = 2x + 1, h = 6x2 + 7x + 2, then fbar = 2x + 1, hbar = x + 2. What is wrong here please ?

The alternative series test says if the absolute value of the terms are not increasing and the limit of the absolute value terms goes to 0, then the alternating series converges. What happens if the converse is satisfied, when the absolute value terms of an alternating series are increasing or the limit of the absolute value terms does not go to zero? If either test fails, can we say the alternating series is divergent?

Can i use one way Anova in SPSS to compare proximate parameters for two samples? For instance, i have two samples subjected to proximate analysis of more than one parameters e.g Ash, Protein, Fat, etc Can i use One way anova to compare siginficant differences of these parameters in the two samples . my sample size is 48 in all? Thanks

If DES weren't present, what was to be used instead? I think that it was some sort of localizations! An advices. UT_PQED.

I am trying to solve the system of ODEs: [math]\frac{d}{dt}(\frac{u}{\sqrt{1-u^2-v^2}})=k_1[/math] [math]\frac{d}{dt}(\frac{v}{\sqrt{1-u^2-v^2}})=k_2[/math] where [math]k_1,k_2[/math] are two constants and [math]u(0)=u_0[/math], [math]v(0)=v_0[/math]. I tried using the symmetry by transforming into polar coordinates: [math]u=\rho(t) cos \theta (t)[/math] [math]v=\rho(t) sin \theta (t)[/math] but it did not help. Any ideas? This is not homework. PS: Wolfram Alpha couldn't solve it even after I made [math]k_1=k_2[/math]

Given the reals a<b find a, t such that : For all r>0 there exist an , x such that : 0<|x-t|<r and a<x<b

Hai all, so in the usual pendulum problem where you have ... well a pendulum swinging from side to side under gravity, you end up with d^2 θ /dt^2 = -g/l sinθ where θ is the angle from the vertical axis, g is acceleration due to gravity, l is the length of the pendulum and t is time. now usually when you solve this, you have to use the small angle approximation on the sinθ so that the eqn turns into d^2 θ /dt^2 = -g/l θ which is rly easy to solve, but my question is, ... before you use the small angle approximation, can you solve the equation analytically and get some kind of function θ(t)? I mean clearly it won't be a trivial solution or even an ea…

Hi, What are ultra-, super-, and sub-diagonal matrices? Thanks Silvia

I am interested in the process and a specific example behind deriving that gamma(x)*gamma(1-x)=pi*csc(pix). The proof or derivation doesn't seem to exist anywhere online.

Suppose I have a function that has some periodic properties related to discontinuities to the left of x=0 and is mostly monotonic or just a continuous to the right of x=0, like for instance y=gamma(x). How do I make such a function and other special functions always periodic over all the domain without reflection symmetry? If I do gamma(-x^2) the function is periodic over the entire domain, but instead of continuing the periodic property from x=-infinity to x=infinity, the function is reflected over the y axis. I want it to be more rotationally symmetric like tan(x).

If you have a flooring or ceiling function somewhere, how do you differentiate it and integrate it?

I had a hunch that since the derivative of the inverse of the function is the reciprocated derivative in the original function as a function of the inverse that, if you had a function that you didn't know the inverse precisely, you could calculate the taylor series for its inverse using that derivative property. Upon researching this, the only conclusive thing I found was something called the Lagrange Inversion theorem. Coincidentally, the way it's described makes it seem like it does exactly what I want it to do, finding the inverse of a function which can only be described analytically. However, I'm not good at reading mathematics from scratch and whenever I look at the…

Sometimes when I have complicated equations involving trig functions I ask wolfram for a solution and it spits out a bunch of imaginary exponents. My guess is it has something to do with the taylor series for e^(i*pi*x) equals cos(x)-isin(x) and e^(-pi*i)=-1, and even in third semester calculus I still haven't seen anything where I solve equations using that identity. But, sometimes I don't want to solve for a number, but rather a formula, and I don't always have wolfram around. So how do I solve complicated trig equations by using Euler's identity? Like let's say I have sqrt(sin(2x)^2+cos(5x)^3)=4. How do I solve that with imaginary/complex exponents?

Sure. Two posts. Show your work. * * a) please provide acceptable standardised thermonuclear test confines. Standard drop height, velocity at detonation, fuel density at centre of momentum, energy released to the atmosphere and any factors involved I've missed that you believe are involved. b) if you could provide me with the material you accept as authoritive on the weak and strong forces I will be able to identify any further equation construction questions I will require guidance on.

I need the Laplace Transformation of Powers of trigonometric functions like the one below ;