Identity Definition and 1000 Threads

For a set S, there is an identity element e with respect to operation * such that for an element a in S: a*e = e*a = a. For a matrix B that is m x n, the identity element for matrix multiplication e = I should satisfy IB = BI = B. But for IB, I is m x m, whereas for BI, I is n x n. Doesn't...

I have been trying to familiarize myself with a particular system of first-order logic with identity, in which the process of substitution is achieved by replacing, one at a time, one occurrence of a variable with a term. (see axiom schemes 6) and 7)). I want to use these axioms to prove the...

1. Homework Statement [/b] f _{a} (z) is defined as f(z) = 1 + az + \frac{a(a-1)}{2!}z^{2}+...+\frac{a(a-1)(a-2)...(a-n+1)}{n!}z^{n} + ... where a is constant Show that for any a,b f _{a+b} (z)= f _{a}(z)f _{b}(z) Homework EquationsThe Attempt at a Solution I've tried starting directly...

Homework Statement http://carlodm.com/images/m.png Homework Equations sin2 et + cos2 et = 1 The identity above is foreign to me. Can anyone explain/have external links that explains this identity? I haven't seen anything like it and Google isn't showing anything useful. Thanks

Homework Statement Show that if p\in (1,\infty) the identity functions id:C^{0}_{1}[a,b]\longrightarrow C^{0}_{p}[a,b] and id:C^{0}_{p}[a,b]\longrightarrow C^{0}_{\infty}[a,b] are not continuous. Homework Equations C^{0}_{p}[a,b] is the space of continuous functions on the [a,b] with...

1. Find the limit of [Cos(x+h)-Cos(x)]/h as h approaches 0 2. Solve using trig identity cos(A+B)= cos(A)cos(B)-sin(A)sin(B) 3. My first class using the actual definition of a derivative. My high school teacher just showed us the shorthand and said "good luck when you get to...

Homework Statement Regarding the identity Ax(BxC) Homework Equations Does this identity only hold when A != B != C?

I noticed that the graphs of sin(x) and sin(x) ^ 2 are very similar. So I offset sin(x) ^ 2 to exactly match sin(x): sin(x) = 2 sin^{2}\left(\frac{x}{2} +\frac{\pi}{4}\right) - 1 Is this right, or is it an illusion? I haven't been able to find any identity that this is based on. If it is...

Is there a name for studying a Lie "group" that doesn't use the identity matrix as a member of the group? I know it's not technically a group anymore, but is there any mathematical work pertaining to the general idea... and what is the terminology so that I can research it better?

Homework Statement div(grad f x grad g)=0. I need to prove this somehow. Homework Equations The Attempt at a Solution I don't really know where to even start this at >.< any help is greatly appreciated.

Homework Statement Derive sin^2 + cos^2 = 1 Homework Equations Use cos 0 =1, cos (x+y) = cos x cos y - sin x sin y Earlier someone posted this same question, but I still don't understand it so please help

1. cos(2x+y) = cos(a+b) = sinasinb + cosacosb = (2x+y) = sin2xsiny + cos2xcosy = sin2x = 2sinxcosx and cos2x = cos2x-sin2x = cos(2x+y) = (cos2x-sin2x)cosy - (2sinxcosx and cos2x)siny (is this correct) 2. evaluate the following exactly (use \sqrt{} in your answer where necessary) cos...

I have been reading papers for my research and I came across this equation twice: \lim_{\eta\to 0+}\frac{1}{x+i \eta} = P\left(\frac{1}{x}\right) - i \pi \delta(x) Where P is the pricipal part. It has been quite a while since I have had complex variables, but might it come from the...

Homework Statement I do not know if the following is correct;if it is,I will be able to save some calculation while doing a problem.Can you please let me know if it is true: \epsilon_{ijk}\*\epsilon_{lmn} = \left(\begin{array}{ccc}\ g_\ {11}&\ g_\ {21}&\ g_\ {31}\\ g_\ {12}&\ g_\...

Two sets are equal iff they contain the same elements. I would argue that two sets that have the same elements are identical as well as equal and that there is a difference between identity and equality. In general {2,3}={3,2} if neither set is defined to be ordered. However obviously {5} \neq...

I was doing a signals and systems problem and I think I might be screwing something up with the cosine function because I get cos(a+b) = cos(a)*cos(b) This is how cos(a+b)=Re\left\{ e^{j*(a+b)} \right\} =Re\left\{ e^{j*(a)}*e^{j*(b)} \right\} =cos(a)*cos(b) Can anyone point out my mistake...

I found this equation last night on Wolfram: http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/06/03/0001/ How is it possible this equation has no restrictions given that the gamma function has poles at the negative integers? Also, won't the zeta function portion run...

If I have a semigroup S, is it possible to partition the set of element S into two semigroups S_1 and S_2 (with S_1 \cap S_2 = 0), in such a way that S_1 has an identity element but S_2 has none?

I forget how this one goes. A cos(x) + B sin (x) = C sin (x + invtan(?)) How do you go about condensing both these terms into 1 like the above?

Homework Statement Prove this is an identity: sin2(x)-sin2(x)=sin(x+y)sin(x-y) Homework Equations N/A The Attempt at a Solution I have made a lot of attempts but can not get one side to equal the other. I know It's something really simple I am missing, but can't figure it out.

From Euler's identity: i^i=exp(-pi/2)= 0.2079 (rounded). I've always thought of this as an interesting result although I don't know of any particular significance or consequence of it. Is there any?

Homework Statement cos^2x-cotx --------------- = cot^2x sin^2x-tanx Homework Equations The Attempt at a Solution every solution I get gives me a zero, not cot^2

Is it true that if \sigma \in S_n is a cycle of length k \leq n, then \sigma^k = \varepsilon, where \varepsilon is the identity permutation, and that k is the least nonzero integer having this property?

Homework Statement I'm finally starting to understand proving trig identities, but I have just one more that I can't seem to figure out. secx - tanxsinx = cosx Homework Equations N/A The Attempt at a Solution Well first, I multiplied the tanx and sinx and came up with sin2x /...

0.15348=0.1415cosβ -0.291sinβcosβ how do i solve this equation with both sinβ and cosβ, i realize that i need to play with the identities but have had no luck, please help i tried squaring the whole thing, and saying cos2β=t, then i get 0.023556=0.02t - 0.582\sqrt{t-1}\sqrt{t}...

Prove this using this identity: k\binom{n}{k}=n\binom{n-1}{k-1} \binom{n}{1}-2\binom{n}{2}+3\binom{n}{3}+...+(-1)n-1\binom{n}{n} I was able to do this via differentiation, but not using this substitution. Any hints would be great.

\sumk=0n\binom{n}{k}2=\binom{2n}{n} Could someone give me a hint as to how to start this. I'm not sure how to really interpret it. (n-k)\binom{n}{k}=n\binom{n-1}{k} Right Side: Suppose you create a committe from \binom{n}{k} , then to pick a leader who isn't in the committee but...

Homework Statement (1-cosx)/(1+cosx) = (cscx-cotx)^2 Homework Equations cscx = 1/sinx, cotx = cosx/sinx, sin^2x + cos^2x = 1 The Attempt at a Solution I have tried many different attempts, but I can't seem to make one side like the other. I took the (cscx-cotx)^2 and expanded it...

Homework Statement solve 3cos(x) + 3 = 2 sin^2(x) where 0 <= x < 2piHomework Equations The Attempt at a Solution 3(cos(x) + 1) = 2 sin^2(x) 3(cos(x) + 1) = 2 (1- cos^2(x)) I've tried this variation, and a couple others but it just does not pan out. Please help. Oh yeah we have a real...

\sum_{m=k}^{n-k}\binom{m}{k}\binom{n-m}{k}=\binom{n+1}{2k+1} I'm not sure how to prove it, I understand the combinatorial proof..i.e. putting it to an example...but i can't derive one side and get the other.

i do not remember the webpage i watched this but i remember that they said ' IN chapter 1 of his notebook Ramanujan wrote ' \sum_{n=0}^{x}n^{r}= (r+1)^{-1}x^{r+1}+ \zeta (-r) - \sum_{k}B_{2k}\frac{\Gamma (r+1)}{\Gamma (k-2r+2)} does anyone knows how to get this ??

Help please on trig identity Homework Statement Simplify cot2xsecx + 1/cosx The Attempt at a Solution Well so far i got: cot2xsecx + 1/cosx =(cos2x/sin2x)(1/cosx) + 1/cosx =((1+cos2x)/(1-cos2x))(1/cosx) + 1/cosx and from there I am stuck, I've tried playing around with it...

Homework Statement Prove Lagrange's identity for real numbers http://mathworld.wolfram.com/LagrangesIdentity.html The Attempt at a Solution I tried one of the methods used in proving the Cauchy-Schwarz inequality (Ax^2 + Bx + C is greater than or equal to zero, where a = the sum from...

There are two things about this identity that I don't understand: 1. Why is it equivalent to a statement of charge conservation? 2. Wikipedia claims that it is like a quantum version of the classical noether's theorem. In what sense is this true? Thanks

Homework Statement Must be proven algebraically, duh! Homework Equations trig identities The Attempt at a Solution I'm at a loss as what to do next. Any help would be appreciated.

Homework Statement I'm trying to find what a a left and right identity element is. Also, I want to see if a one sided element for * exists, if it is unique. Homework Equations The Attempt at a Solution Ok, I just don't really know what a one sided element is. I'm using...

Let X and X' denote a single set in the two topologies T and T', respectively. Let i:X'-> X be the identity function a. Show that i is continuous <=> T' is finer than T. b. Show that i is a homeomorphism <=> T'=T This is all I've got. According to the first statement... X \subset T and...

Hi, I read the chapter "Anticommuting Numbers" by Peskin & Schröder (page 299) about Grassmann Numbers and now I would like to prove \int d \bar{\theta}_1 d \theta_1 ... d \bar{\theta}_N d \theta_N e^{-\bar{\theta} A \theta} = det A \theta_i are complex Grassmann Numbers...

Homework Statement Prove: csc2@= 1/(1-(sin@-cos@)^2 Homework Equations The Attempt at a Solution I'm stuck can't seem to work this on out. I'm not seeing the relationship between the two

Homework Statement Find the exact value of the expression Tan(3/4-12/5) Homework Equations tan(x-y)= (tan3/4 -tan12/5)/ (1+tan3/4tan12/5) The Attempt at a Solution I am not sure how to get exact values for these ratios. I haven't been able to get past this point

Homework Statement \bigtriangledown\times\\(v\times w)= v(\bigtriangledown\cdot w) - w(\bigtriangledown\cdot v)+ (v\cdot\bigtriangledown)w - (w\cdot\bigtriangledown) v I've tried expanding left side and get [v1(dw2/dy+dw3/dz)-w1(dv2/dy+dv3/dz)]i +...

Hi Everyone, Do there exist any explicit formula for Cos(x_1+x_2+...+x_n) as a sum of products of Sin(x_i) & Cos(x_i)? Or we need to expand using Cos(A+B), Sin(A+B) again & again? If it exists then what is about Sin(x_1+x_2+...+x_n)? [It is understood that there will be 2^(n-1) number of...

Homework Statement Differentiate the identity sin2x = 2sinxcosx to develop the identity for cos2x, in terms on sin x and cos x Homework Equations The Attempt at a Solution Im not sure where to start with this one. Should I find the derivative of both sides of the equation, and...

Homework Statement Show that (S, *) is a group where S is the set of all real numbers except for -1. Define * on S by a*b=a+b+ab The Attempt at a Solution Well I know that i have to follow the axioms to prove this. So I started with G1 which is associativity. This one I got to...

Homework Statement If A = I + uv*, where u and v are m vectors and A is known to be nonsingular, show that the inverse of A = I + \alphauv* where \alpha is a scalar value Homework Equations The Attempt at a Solution Since A is nonsingular, we know the rank of A is m. Since both...

Homework Statement Given S is a Skew-Hermitian (S*=-S), Prove that I - S is a nonsingular matrix Homework Equations If a matrix A is nonsingular, for Ax=0, x={0} The Attempt at a Solution (I-S)x=0, and I have been trying to show that the solution for x is always zero. Is this the...

I have seen the following identity used. Exp[iw/2]-Exp[iw/2]=Exp[iw]-1 I can't find this in any book and I can't prove it myself. The left side equals 2isin(w/2) The right side equals cos(w)+isin(w)-1 On the face of it, that seems to make the identity absurd How can one go about proving...

Hi everyone, Any assistance with this following problem would be greatly appreciated. I'm in Year 11 and working through Apostol volume 1. Homework Statement sin n*pi = 0, where n is an integer sin n*pi =/= 0, where n is not an integer. Prove these statements... Homework...

Homework Statement Basically I am finishing of a projectile question and I get stuck here: Trying to find \theta \frac{1}{2} (sin2 \theta) tan^2 \theta -tan \theta + \frac{1}{2} sin2 \theta = 0 Homework Equations The Attempt at a Solution I tryed spliting tan \theta into \frac{sin...

Homework Statement sinx cos2x=1 x is greater than or equal to 0 and less than 2pi Homework Equations What I used: 1-2sin2x cos2x-sin2x but there might be one that I didn't and should have... The Attempt at a Solution Basically I have...