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eigenfunction

ˈeigenfunction Physics. [tr. G. eigenfunktion.] A solution of a differential equation possessing solutions only for special values of a parameter.1926 Proc. R. Soc. A. CXII. 661 A set of independent solutions, which may be called eigenfunctions. 1927 [see eigenvalue]. 1938 Nature 16 Apr. 668/1 The h... Oxford English Dictionary

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Eigenfunction

An eigenfunction is a type of eigenvector. If f(t) is an eigenfunction of D with eigenvalue λ, then Ab = λb. wikipedia.org

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Eigenfunction in functional calculus Let $X$ be a complex Banach space, $A\in L(X)$ and $F$ be an analytic function in a neighborhood of $\sigma(A)$. Now I want to show that if $x\in X$ is an eigenfunction of $A$ corr...

Let $F(z) = \sum_{n=0}^\infty a_n (z-\lambda_0)^n$, with $\lambda$ inside the radius of convergence (you could just take $\lambda=\lambda_0$). Then $$ \begin{split} F(A)x &= \left(\sum_{n=0}^\infty a_n (A-\lambda_0 I)^n x\right) \\\ &= \sum_{n=0}^\infty a_n (A-\lambda_0 I)^n x = \sum_{n=0}^\infty a_...

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Eigenfunction and their orthogonality with respect to the weight function The Eigenfunction and their orthogonality with respect to the weight function $\sigma$ is defined as $$\int_{a}^{b}\phi _n\text{(x)}\phi _m\tex...

In which case, the $\tanh(\ldots)\sinh(\ldots)$ are the riven functions, since they are orthonormal, taking an inner product with the $m$-th eigenfunction

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How are eigenfunctions found? Let's stick to the derivative operator and any other operators built up from it. For instance, if $D = \frac{d}{dt}$, then the eigenfunction of $D$ is known to be $e^{at}$, associated wit...

For a minimal example an eigenfunction to the operator ${\bf T = D}^2+2{\bf D}$ on the space of $\\{\sin(x),\cos(x)\\}$.

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Eigenfunctions of a second derivative operator Consider the operator $L :=\frac{-d^2}{dy^2}+ \alpha^2 - K(y)$ on the space of functions $f(y) $ on $H^2(-a,a) \cap H_0^1(-a,a)$. Here $K(y)$ is an even function and $\al...

If $f$ satisfies, $$ -f''+a^2f-K(y)f=\lambda f, \quad f(-1)=f(1)=0. $$ then so do $$ f_{even}=\frac{1}{2}\big(f(x)+f(-x)\big), \quad f_{odd}=\frac{1}{2}\big(f(x)-f(-x)\big). $$ Hence, the eigenspace of our operator is spanned by odd and even eigenfunctions. We shall next show that only one of the tw...

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Eigenfunction of all Hecke operators Let $f$ be a cusp form in $S_{16}$. I want to show $f$ is an eigenfunction of all Hecke operators, i.e., $T_n(f)=\lambda_nf$. I know Eisenstien series are eigenfunctions of all H...

$$S_{16}(SL_2(\Bbb{Z}))= \Delta\ M_4(SL_2(\Bbb{Z}))$$ is of dimension $1$

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First Eigenfunction of Simple Equation Consider the interval $[-a,a]$ and the following problem: $$\phi'' + \lambda\phi=0$$ $$ \phi(\pm a) = 0. $$ The obvious sequence of orthogonal eigenfunctions seems to be $\sin(...

2a}x\right) ~~ \text{for odd } n = 1,3,5,\ldots$$ $$\sin\left(\frac{n\pi}{2a}x\right) ~~ \text{for even } n = 2,4,6,\ldots$$ As you can see, the first eigenfunction

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Eigenfunction expansion solution to a PDE with a constant non homogeneous term I'm wondering if the method of finding a solution to a nonhomogeneous PDE by the method of eigenfunction expansion works if the nonhomogen...

An infinite series of sines can indeed converge to a constant, in the interior of the interval in question (I guess you have Dirichlet boundary conditions at the endpoints). If you look at the sum of that Fourier series on the whole real line you will see a square wave, where the part that you're in...

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Show that the function is an eigenfunction of the equation I'm not sure how to use the bbcode so I've taken a screenshot instead: !enter image description here Came up on a past exam paper that I'm working towards ...

$$(\sin\lambda_nx)'=\lambda_n\cos\lambda_nx$$ $$(\sin\lambda_nx)'=-\lambda_n^2\sin\lambda_nx$$ So putting $\,\psi(x):=\sin\lambda_nx\,$ , we easily find the above are solutions to the given differential equation, and in order to have $\,\psi(1)=0\,$ we must choose $\,\lambda_n=k_n\pi\,\,,\,\,k_n\in\...

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The choice of the eigenfunction of Laplacian $M$ is a closed Riemannian manifold and $\lambda_1>0$ is the first nontrivial eigenvalue of $\Delta$. Can we find a eigenfunction $f$ of $\lambda_1$ such that $\mathop {\su...

Let $f$ be a eigenfunction of $\Delta$ with respect to $\lambda_1$.

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Value of the Eigenfunction at a point I'm reading "Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation" < At a certain point the author states "where φi(p) is the value of the eigenfunction...

Note that it's an _eigenfunction_ not eigenvector. For example an eigenfunction with eigenvalue $\lambda$ of the operator $$\frac{d}{dx}$$ is a solution to $$\frac{d}{dx}f=\lambda f$$ which is $f= e^{\lambda

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How to find the corresponding eigenfunction after determining the eigenvalues? I was reading this page (< example 4.1.4, which says: > Again $A$ cannot be zero if $\lambda$ is to be an eigenvalue, and $sin(\sqrt {\la...

The corresponding eigenfunction is then the result of plugging in the corresponding eigenvalue: $x_k(t) = \cos{k t}$. Note we ignore the constant $A$ here for now; the coefficient of an eigenfunction will be determined elsewhere.

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How to show that λ is an eigenvalue of D by finding a corresponding eigenfunction Define the _differentiation operator_ $D$ to be the transformation $D : C_\infty(\mathbb R)\to C_\infty(\mathbb R)$ given by $D( f ) = ...

**Hint:** For a given number $\lambda$, consider $f(x)=e^{\lambda x}$, which is a vector in your vector space. Why is this an eigenvector? What is the associated eigenvalue?

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Transform the integral equation into a Sturm-Liouville problem using eigenfunction expansions Transform the integral equation $\int_0^1 k(x,y)u(y)dy-\lambda u(x)=x$ into a Sturm-Liouville problem using eigenfunction e...

You can recognise the function $k(x,y)$ as the Green's function for the operator $L = \frac{\text{d}^2}{\text{d} x^2}$ on the domain $[0,1]$ with homogeneous Dirichlet boundary conditions. If you now would be given the inhomogeneous differential equation $L v = f$, then the solution to that equation...

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eigenfunction

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eigenfunction

Eigenfunction

Eigenfunction in functional calculus Let $X$ be a complex Banach space, $A\in L(X)$ and $F$ be an analytic function in a neighborhood of $\sigma(A)$. Now I want to show that if $x\in X$ is an eigenfunction of $A$ corr...

Eigenfunction and their orthogonality with respect to the weight function The Eigenfunction and their orthogonality with respect to the weight function $\sigma$ is defined as $$\int_{a}^{b}\phi _n\text{(x)}\phi _m\tex...

How are eigenfunctions found? Let's stick to the derivative operator and any other operators built up from it. For instance, if $D = \frac{d}{dt}$, then the eigenfunction of $D$ is known to be $e^{at}$, associated wit...

Eigenfunctions of a second derivative operator Consider the operator $L :=\frac{-d^2}{dy^2}+ \alpha^2 - K(y)$ on the space of functions $f(y) $ on $H^2(-a,a) \cap H_0^1(-a,a)$. Here $K(y)$ is an even function and $\al...

Eigenfunction of all Hecke operators Let $f$ be a cusp form in $S_{16}$. I want to show $f$ is an eigenfunction of all Hecke operators, i.e., $T_n(f)=\lambda_nf$. I know Eisenstien series are eigenfunctions of all H...

First Eigenfunction of Simple Equation Consider the interval $[-a,a]$ and the following problem: $$\phi'' + \lambda\phi=0$$ $$ \phi(\pm a) = 0. $$ The obvious sequence of orthogonal eigenfunctions seems to be $\sin(...

Eigenfunction expansion solution to a PDE with a constant non homogeneous term I'm wondering if the method of finding a solution to a nonhomogeneous PDE by the method of eigenfunction expansion works if the nonhomogen...

Show that the function is an eigenfunction of the equation I'm not sure how to use the bbcode so I've taken a screenshot instead: !enter image description here Came up on a past exam paper that I'm working towards ...

The choice of the eigenfunction of Laplacian $M$ is a closed Riemannian manifold and $\lambda_1>0$ is the first nontrivial eigenvalue of $\Delta$. Can we find a eigenfunction $f$ of $\lambda_1$ such that $\mathop {\su...

Value of the Eigenfunction at a point I'm reading "Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation" < At a certain point the author states "where φi(p) is the value of the eigenfunction...

How to find the corresponding eigenfunction after determining the eigenvalues? I was reading this page (< example 4.1.4, which says: > Again $A$ cannot be zero if $\lambda$ is to be an eigenvalue, and $sin(\sqrt {\la...

How to show that λ is an eigenvalue of D by finding a corresponding eigenfunction Define the _differentiation operator_ $D$ to be the transformation $D : C_\infty(\mathbb R)\to C_\infty(\mathbb R)$ given by $D( f ) = ...

Transform the integral equation into a Sturm-Liouville problem using eigenfunction expansions Transform the integral equation $\int_0^1 k(x,y)u(y)dy-\lambda u(x)=x$ into a Sturm-Liouville problem using eigenfunction e...