Identity Definition and 1000 Threads

Given that 1/ f(cx) = k - g(x) and 2/ the above is an identity, where f(.) and g(.) are two functions and c, k are real valued constants. The problem is to infer upon the types of f(.) and g(.). I have a hunch that f(.) and g(.) are logarithimic functions. Can anyone provide...

for every sequence of numbers a_n E_n is this identity correct ? \sum_{n= -\infty}^{\infty}a_n e^{2\pi i E_{n}}= \sum_{n= -\infty}^{\infty}a_n \delta (x-E_{n})

Suppose we have a statement A that holds if and only if statement B holds. "A if and only if B" I'm fairly sure I read before that this does not necessarily mean that A and B are identical: in general, A <--> B does not imply A = B. I'm having difficulty determining how A and B could be...

Suppose \phi is a scalar function: R^n\to R, and it satisfies the Poisson equation: \nabla^2 \phi=-\dfrac{\rho}{\varepsilon_0} Now I want to calculate the following integral: \int \phi \nabla^2 \phi \,dV So using Greens first identity I get: \int \phi \nabla^2 \phi \,dV = \oint_S \phi...

Alright, I'll need some help formulating this, since my writing tends to be... well... just not very eloquent and representative of my thoughts. I don't believe in soul, afterlife, or other nonsense. I think our self, our consciousness, is a function of our complex brains. For what...

Homework Statement Let the domain D be bounded by the surface S as in the divergence theorem, and assume that all fields satisfy the appropriate differentiability conditions. Suppose that: \nabla\cdot\vec{V}=0 \vec{W}=\nabla\phi with \phi = 0 on S prove...

Hey folks, I'm reading the paper: http://arxiv.org/abs/hep-ph/0301168 and I'm trying to make sense of the first line of eqtn 44 where he states that we can write: \frac{1}{2}\sum \int\frac{d^{2n}k}{(2\pi)^{2n}}log(k^2+\frac{m^2}{L^2}) as...

I am having difficulty symbolically resolving the LHS of this identity algebraically: \frac{r}{2} \left[ \left(8 \pi P(r) + \frac{1}{r^2} \right) \frac{r}{r - 2u} - \frac{1}{r^2} \right] = \frac{4 \pi r^3 P(r) + u}{r(r - 2u)} \left(4 \pi r P(r) + \frac{1}{2r} \right) \frac{r}{r - 2u} -...

I was playing around with Euler's identity the other day. I came across something that seems contradictory to everything else I know, but I can't really explain it. I started with e^{i\pi} = -1. I rewrote this as ln[-1] = i\pi Multiplying by a constant, we have kln[-1] = ki\pi...

Hi, I'm new to this site and I'm very happy that I found it. My Pre-cal 2 teacher has been no help to me when it comes to explaining certain steps needed to solve a problem. Overall I'm having a hard time choosing the correct identity needed to solve the problems. However what I do not...

Homework Statement Prove that there is no holomorphic function f in the open unit disk such that f(1/n)=((-1)^n)/(n^2) for n=2,3,4... Homework Equations The identity theorem: Let f and g be holomorphic functions in the connected open subset of C, G. If f(z)=g(z) for all z in a subset...

Homework Statement Suppose P \in L(V) and P^2 = P. Prove that (I+P) is invertible. Homework Equations The Attempt at a Solution Am I right to assume that since P^2 = P, P = I?

well trying to help my little brother with some chem homework, and i believe i am just thinking too hard about this question anyways its a whole chem lab thingy about the composition of pennies and one of the thinking questions is now when it says "identity" I am assuming they are...

[SOLVED] Identity in a subring Homework Statement In Dummit & Foote on the section on tensor product of modules (10.4 pp.359), the authors write "Suppose that the ring R is a subring of the ring S. Throughout this section, we always assume that 1_R=1_S (this ensures that S is a unital...

[SOLVED] Sigma Notation Question/Trig Identity I posted this elsewhere but I think I put it in the wrong place so I'm going to post my question again here. Basically I have to deduce the second formula from the first. Both equations are the same except for the top of the right side...

Homework Statement The identity below is significant because it relates 3 different kinds of products: a cross product and a dot product of 2 vectors on the left side, and the product of 2 real numbers on the right side. Prove the identity below. | a × b |² + (a • b)² = |a|²|b|²...

Has anyone seen this identity: g^{ab}\nabla g_{ab}=\nabla ln|g| I've seen it used, but want to figure out where it comes from. Does anyone know a name or have any ideas??

Integral inequality and comparison Homework Statement Prove the inequality \frac{2}{(n+1) \cdot \pi} \leq a_n \leq \frac{2}{n \pi}} where a_n = \int_{0}^{\pi} \frac{sin(x)}{n \cdot \pi +x} dx and n \geq 1 The Attempt at a Solution Proof: If n increased the left side of...

The question is to prove for finite dimensional T: V to W, T is injective iff there exists an S: W to V such that ST is the identity map on V. I can't quite make the connection between injectivity and the identity map. any suggestions? thanks in advance.

Homework Statement derive the identity: del((F)^2) = 2 F . del(F) + 2Fx (del x F) the dot is a dot product Homework Equations The Attempt at a Solution first i set F = <a,b,c>, making F^2 = a^2 + b^2 + c^2 I took the partial derivatives with respect to x, y, and z (to get the necessary parts...

Will some one help me to prove this identity G(n)+G(1-n)= pi/ sin npi 0<n<1 B(m,n) = (m-1)! / n(n+1)...(n+m+1) ,for beta function

Homework Statement Need to prove these 2 identities of beta function & gamma function ? Homework Equations G(n)G(1-n)= pi/sin npi B(m,n) = (m-1)! / n (n+1)...(n+m+1) The Attempt at a Solution I tired using beta function in 1st one but did not get the solution .

Homework Statement Prove that \nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - (\nabla \cdot \nabla)\vec{A} using Einstein notation. Homework Equations \nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - (\nabla \cdot \nabla)\vec{A}...

Help with a Trigonometric identity... Homework Statement (sin x + sin 2x + sin 4x) / (cos x + cos 2x + cos 4x) = tan 2x Homework Equations sin 2x = 2sinxcosx; cos 2x = cos^2x - sin^2x The Attempt at a Solution solving left side, =[sin x + sin 2x + sin (2x + 2x)]/[cos x + cos...

Homework Statement Prove that the Given Equation is an Identity: sin2A ------ = cotA 1 - cos2A Homework Equations sin(A+B) = sinAcosB + cosAsinB cos(A+B) = cosAcosB - sinAsinB tan(A+B) = (tanA + tanB) / (1 - tanAtanB) sin2A = 2sinAcosA cos2A = cos^{}2A - sin^{}2A...

Hi I'm having trouble understanding how they simplified this integration using trig substitutions...I really don't know what identities they used to make these substitutions or the strategy behind why this substitution was made(particularly the 3 steps in the red box). Obviously it works but I...

More Trig Identity Proofs ... Homework Statement 1. cot^2x - 1 = cot2x ----------- 2cotx2. tanx + cotx = 2csc2x3. cos(A+B) = 1-tanAtanB --------- ---------- cos(A-B) 1+tanAtanB Homework Equations The Attempt at a Solution...

[SOLVED] trig identity Homework Statement Can someone help me prove that \sum_{k=1}^{(n-1)/2}\cos(2 \pi k / n) = -1/2 where n is an odd number?Homework Equations The Attempt at a Solution I don't know where to start. You can easily verify it is true for n=3. But after that things get...

Homework Statement My professor stated the theorem "If (X,<,>) is an an inner product space and || || is the norm generated by <,>, then we have ||x+y||² + ||x-y||² = 2(||x||² + ||y||²)." But then she also said that the converse was true. I suppose this means that "Given (X, || ||) a normed...

Homework Statement cos^2x - cos^4x = cos^2x sin^2x Homework Equations N/A The Attempt at a Solution L.S. = cos^2x -(cos^2x)(cos^2x) = cos^2x -cos^2x(cos^2x) I'm stuck here. Am I doing something wrong during the first step? Thanks for your help.

Homework Statement 1. (sinx - cosx)(sinx + cosx) = 2sin^2x-1 2. (2sinx + 3cos)^2 + (3sinx - 2cosx)^2 = 13 Homework Equations N/A The Attempt at a Solution For 1. L.S. = sinx^2+sinxcosx-sinxcosx-cosx^2, the sinxcosx cancels and I'm lost. I haven't a clue how to do the second...

Hi This is one of the problems for my take home final exam on differential equations. I have been looking for a solution for this problem intensely for the last two days. This problem comes from Calculus vol 2 by Apostol section 6.24 ex 7. here it is Homework Statement Use the identities...

How can we proceed to prove the following identity ?

Homework Statement Prove that the following binomial identity holds: {n+k-1 \choose k} = \sum_{i=1}^k {k-1\choose i-1}{n\choose i} The Attempt at a Solution One of the methods I've tried is to use induction on the variable n, but while trying this I got stuk on rewriting the...

Homework Statement start from: x = [x' + vt']/sqrt[1 - v^2/c^2] ct = [v/cx' +ct']/sqrt[1 - v^2/c^2] y = y' z = z' Homework Equations show that ( 1 - \frac{u^{2}}{c^{2}})(1+\frac{vux'^{2}}{c^{2}}) = ( 1 - \frac{v^{2}}{c^{2}})(1-\frac{u'^{2}}{c^{2}}) The Attempt at a Solution...

Hi. The problem is as follows: Homework Statement Let m and n be integers, we may assume that (if they are not equal), m is the smallest. Then \sum_{i=0}^m \sum _{j=0}^n f((m+n)-2 (i+j)) = \sum_{i=0}^m \sum _{j=0}^{-2 i+m+n} f((m+n)-2 (i+j)) for some sequence f(k)_k. Homework Equations...

Homework Statement (1) (1+cosx)/(sinx)= cot (x/2) (2) 2 csc 2x= sec x csc x (3) cos^6 x- sin^6 x= cos 2x(1 - 1/4sin^2 2x) ( I think this has to do something with subtracting -3a^2b^2, since I need to get a-2ab+b to factor it..?) Homework Equations Addition and Subtraction...

Homework Statement Show that \prod_{k=1}^{n-1}\sin\frac{k\pi}{2n}=\frac{\sqrt{n}}{2^{n-1}} The Attempt at a Solution I have no idea where to start.

[SOLVED] Trig Identity Homework Statement cos^4 (x) = (3/8) + (1/2)(cos(2x)) + (1/8)(cos(4x)) Homework Equations cos2x = 2cos^2 x - 1 cos^2 x = 1 - sin^2 x The Attempt at a Solution Can someone please give me hints? Thanks.

Hello all. While going back to group theory basics to make sure i understand rather than just know the fundamentals i came across for the first time ( having read many books ) the weak versus strong versions of the identity axiom. The strong version says that a group must have a unique...

I have a trouble proving that a finate (nonzero) commutative ring with no zero divisors must have an identity with respect to multiplication. Could anybody please give me some hints? I do know all the definitions (of ring, commutative ring, zero divisors, identity) but have no idea how to go...

I derived the following identity after considering my thread "Recursive series equality" but the result is so clean and neat that I post it as a new topic. Let S_{0} = 0 \quad S_{1} = 1 \quad S_{n} = b*S_{n-1} - S_{n-2} Then S_{n}*(S_{n+b} -S_{b-2}) = (S_{n+1}+1)*(S_{n+b-1} -S_{b-1})...

If z=cis\theta, verify that \tan \theta = \frac{{z - z^{ - 1} }}{{i(z + z^{ - 1} )}} . Use this result to prove that \cos (2\theta ) = \frac{{1 - \tan ^2 \theta }}{{1 + \tan ^2 \theta }} Ok, I've managed to verify the first equation given, but I am not really sure how to use it to...

problem prove that: \forall n \in N \forall 0<=k<=2^{n-1} (C(2^n,k)=\sum_{j=0}^{k}C(2^{n-1},j)C(2^{n-1},k-j)) attempt at solution induction seems to be too long I am opting for a shorter solution, so the sum that it's wrriten in the rhs is the square of the sum of the term C(2^(n-1),j) but...

Homework Statement Prove that there is at most one real number b with the property that br=r for all real numbers r. (Such a number is called a multiplicative identity) Note: to show there is a unique object with a certain property, show that (1) there is an object with the property and (2)...

Dear All Does anyone have an online (preferably) source on the Bianchi identity on p-form fields (dF=0)? I would like to read more on the various cases, particularly the physical meaning of a violated Bianchi identity. Thanks ...

I know that \frac{1-cos(x)}{2sin\left(\frac{x}{2}\right)} = sin\left(\frac{x}{2}\right) but is there a trig identity that states this? I've been manipulating a certain equation to try and fit a trig identity to make everything make sense. Actually, I started out with...

please help me to solve this identity ((\phi\lambda)\chi)+((\lambda\chi)\phi)+((\chi\phi)\lambda)=0 where () = poisson bracket \phi=\phi(t,q_{i},p_{i}) \chi=\chi(t,q_{i},p_{i}) \lambda=\lambda(t,q_{i},p_{i}) for i=1,2,...,n

Homework Statement Prove the identity 11. 1 - co5xcos3x - sin5xsin3x = 2sin^2x 50. ln |secx + tanx| = -ln |secx - tanx| 52. The following equation occurs in the study of mechanics: \sin\theta = \frac{I_1\cos\phi}{\sqrt{(I_1\cos\phi)^2 + (I_2\sin\phi)^2}}. It can happen that I_1 = I_2...

Homework Statement I want to show: \langle x' - \triangle x' \vert \alpha \rangle = \langle x' \vert \alpha \rangle - \triangle x' \dfrac{\partial}{\partial x'}\langle x' \vert \alpha \rangle Homework Equations \vert \alpha \rangle is a state. The Attempt at a Solution i...