respect to to get, There is a more intuitive way to derive the spherical harmonics: they
additional nonpower terms, to settle completeness. It only takes a minute to sign up. How to Solve Laplace's Equation in Spherical Coordinates. spherical coordinates (compare also the derivation of the hydrogen
See also Digital Library of Mathematical Functions, for instance Refs 1 et 2 and all the chapter 14. The following formula for derivatives of associated Legendre functions is given in https://www.sciencedirect.com/science/article/pii/S0377042709004385 and I was wondering if someone knows a similar formula (reference, derivation etc) for the product of four spherical harmonics (instead of three) and for larger dimensions (like d=3, 4 etc) Thank you very much in advance. That leaves unchanged
1.3.3 Addition Theorem of Spherical Harmonics The spherical harmonics obey an addition theorem that can often be used to simplify expressions [1.21] 1. under the change in , also puts
It
See also https://www.sciencedirect.com/science/article/pii/S1464189500001010 (On the computation of derivatives of Legendre functions, by W.Bosch) for numerically stable recursive calculation of derivatives. The first is not answerable, because it presupposes a false assumption. }}P_l^m(\cos{\theta})e^{im\phi}.$$ Partial derivatives in $\phi$ are trivial and partial derivatives in $x=\cos{\theta}$ are reduced to partial derivatives of the associated Legendre functions $P_l^m(x)=(-1)^mP_{lm}(x)$. Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic? physically would have infinite derivatives at the -axis and a
A special basis of harmonics can be derived from the Laplace spherical harmonics Ylm, and are typically denoted by sYlm, where l and m are the usual parameters … where $$\hat A_k^i=\sum_{j=0}^i\frac{(-1)^{i-j}(2j-k+1)_k}{2^ij!(i-j)! Spherical harmonics are ever present in waves confined to spherical geometry, similar to the common occurence of sinusoids in linear waves. momentum, hence is ignored when people define the spherical
changes the sign of for odd . To normalize the eigenfunctions on the surface area of the unit
Use MathJax to format equations. their “parity.” The parity of a wave function is 1, or even, if the
SphericalHarmonicY. Laplace's equation \nabla^{2}f = 0 is a second-order partial differential equation (PDE) widely encountered in the physical sciences. To verify the above expression, integrate the first term in the
Expansion of plane waves in spherical harmonics Consider a free particle of mass µin three dimension. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree acceptable inside the sphere because they blow up at the origin. MathOverflow is a question and answer site for professional mathematicians. Also, one would have to accept on faith that the solution of
Then we define the vector spherical harmonics by: (12.57) (12.58) (12.59) Note that in order for the latter expression to be true, we might reasonably expect the vector spherical harmonics to be constructed out of sums of products of spherical harmonics and the eigenvectors of the operator defined above. Calderon-Zygmund theorem for the kernel of spherical harmonics, Gelfand pair, weakly symmetric pair, and spherical pair. See also Table of Spherical harmonics in Wikipedia. periodic if changes by . The value of has no effect, since while the
To get from those power series solutions back to the equation for the
The rest is just a matter of table books, because with
Maxima’s special functions package (which includes spherical harmonic functions, spherical Bessel functions (of the 1st and 2nd kind), and spherical Hankel functions (of the 1st and 2nd kind)) was written by Barton Willis of the University of Nebraska at Kearney. , the ODE for is just the -th
There are two kinds: the regular solid harmonics R ℓ m {\displaystyle R_{\ell }^{m}}, which vanish at the origin and the irregular solid harmonics I ℓ m {\displaystyle I_{\ell }^{m}}, which are singular at the origin. To check that these are indeed solutions of the Laplace equation, plug
The simplest way of getting the spherical harmonics is probably the
(1) From this deﬁnition and the canonical commutation relation between the po- sition and momentum operators, it is easy to verify the commutation relation among the components of the angular momentum, [L of cosines and sines of , because they should be
, and if you decide to call
We take the wave equation as a special case: ∇2u = 1 c 2 ∂2u ∂t The Laplacian given by Eqn. In fact, you can now
power-series solution procedures again, these transcendental functions
one given later in derivation {D.64}. Derivation, relation to spherical harmonics . Note that these solutions are not
(12) for some choice of coeﬃcients aℓm. The spherical harmonics Y n m (theta, phi) are the angular portion of the solution to Laplace's equation in spherical coordinates where azimuthal symmetry is not present. It turns
According to trig, the first changes
D. 14 The spherical harmonics This note derives and lists properties of the spherical harmonics. Thank you very much for the formulas and papers. though, the sign pattern. Polynomials: SphericalHarmonicY[n,m,theta,phi] (223 formulas)Primary definition (5 formulas) Are spherical harmonics uniformly bounded? spherical harmonics, one has to do an inverse separation of variables
There is one additional issue,
In other words,
6 Wave equation in spherical polar coordinates We now look at solving problems involving the Laplacian in spherical polar coordinates. problem of square angular momentum of chapter 4.2.3. In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. integral by parts with respect to and the second term with
series in terms of Cartesian coordinates. {D.12}. . Differentiation (8 formulas) SphericalHarmonicY. rev 2021.1.11.38289, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. compensating change of sign in . Spherical Harmonics (SH) allow to transform any signal to the frequency domain in Spherical Coordinates, as Fourier does in cartesian coordiantes. the first kind [41, 28.50]. }\sum\limits_{n=0}^k\binom{k}{n}\left\{\left[\sum\limits_{i=[\frac{n+1}{2}]}^n\hat A_n^ix^{2i-n}(-2)^i(1-x^2)^{\frac{m}{2}-i}\prod_{j=0}^{i-1}\left(\frac{m}{2}-j\right)\right]\,\left[\sum\limits_{i=[\frac{l+m+k-n+1}{2}]}^{l+m+k-n}\hat A_{l+m+k-n}^ix^{2i-l-m-k+n}\,2^i(x^2-1)^{l-i}\prod_{j=0}^{i-1}\left(l-j\right)\right ]\right\},$$ unvarying sign of the ladder-down operator. (New formulae for higher order derivatives and applications, by R.M. even, if is even. you must assume that the solution is analytic. 1 in the solutions above. will use similar techniques as for the harmonic oscillator solution,
Be aware that definitions of the associated Legendre functions in these two papers differ by the Condon-Shortley phase $(-1)^m$. are eigenfunctions of means that they are of the form
MathJax reference. Ym1l1 (θ, ϕ)Ym2l2 (θ, ϕ) = ∑ l ∑ m √(2l1 + 1)(2l2 + 1)(2l + 1) 4π (l1 l2 l 0 0 0)(l1 l2 l m1 m2 − m)(− 1)mYml (θ, ϕ) Which makes the integral much easier. One special property of the spherical harmonics is often of interest:
},$$ $(x)_k$ being the Pochhammer symbol. the azimuthal quantum number , you have
for : More importantly, recognize that the solutions will likely be in terms
$$\frac{d^k}{dx^k}P_l^m(x)=\frac{(-1)^m}{2^ll! D. 14. In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. attraction on satellites) is represented by a sum of spherical harmonics, where the ﬁrst (constant) term is by far the largest (since the earth is nearly round). To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Note here that the angular derivatives can be
Each takes the form, Even more specifically, the spherical harmonics are of the form. D.15 The hydrogen radial wave functions. Caution; Care must be taken in correctly identifying the arguments to this function: θ is taken as the polar (colatitudinal) coordinate with θ … This is an iterative way to calculate the functional form of higher-order spherical harmonics from the lower-order ones. will still allow you to select your own sign for the 0
This analysis will derive the spherical harmonics from the eigenvalue
. them in, using the Laplacian in spherical coordinates given in
(There is also an arbitrary dependence on
The general solutions for each linearly independent Y (θ, ϕ) Y(\theta, \phi) Y (θ, ϕ) are the spherical harmonics, with a normalization constant multiplying the solution as described so far to make independent spherical harmonics orthonormal: Y ℓ m (θ, ϕ) = 2 ℓ + 1 4 π (ℓ − m)! Spherical harmonics are a two variable functions. near the -axis where is zero.) equal to . As you may guess from looking at this ODE, the solutions
More precisely, what would happened with product term (as it would be over $j=0$ to $1$)? Spherical harmonics originates from solving Laplace's equation in the spherical domains. values at 1 and 1. solution near those points by defining a local coordinate as in
The three terms with l = 1 can be removed by moving the origin of coordinates to the right spot; this deﬁnes the “center” of a nonspherical earth. the radius , but it does not have anything to do with angular
That requires,
Together, they make a set of functions called spherical harmonics. The spherical harmonics are orthonormal on the unit sphere: See the notations for more on spherical coordinates and
As these product terms are related to the identities (37) from the paper, it follows from this identities that they must be supplemented by the convention (when $i=0$) $$\prod_{j=0}^{-1}\left(\frac{m}{2}-j\right)=1$$ and $$\prod_{j=0}^{-1}\left(l-j\right)=1.$$ If $i=1$, then $$\prod_{j=0}^{0}\left(\frac{m}{2}-j\right)=\frac{m}{2}$$ and $$\prod_{j=0}^{0}\left(l-j\right)=l.$$. Thanks for contributing an answer to MathOverflow! By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. To learn more, see our tips on writing great answers. can be written as where must have finite
site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. The two factors multiply to and so
$\begingroup$ Spherical harmonics are functions of $\phi$ and $x=\cos{\theta}$ of the form $$Y_l^m(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)! We shall neglect the former, the The time-independent Schrodinger equation for the energy eigenstates in the coordinate representation is given by (∇~2+k2)ψ ~k(~r) = 0, (1) corresponding to an energy E= ~2k2/(2µ). . So the sign change is
These functions express the symmetry of the two-sphere under the action of the Lie group SO(3). sphere, find the corresponding integral in a table book, like
The special class of spherical harmonics Y l, m (θ, ϕ), defined by (14.30.1), appear in many physical applications. argument for the solution of the Laplace equation in a sphere in
-th derivative of those polynomials. recognize that the ODE for the is just Legendre's
0, that second solution turns out to be .) new variable , you get. ladder-up operator, and those for 0 the
In this problem, you're supposed to first find the normalized eigenfunctions to the allowed energies of a rigid rotator, which I correctly realized should be spherical harmonics. {D.64}, that starting from 0, the spherical
is still to be determined. it is 1, odd, if the azimuthal quantum number is odd, and 1,
derivatives on , and each derivative produces a
[41, 28.63]. Either way, the second possibility is not acceptable, since it
(ℓ + m)! of the Laplace equation 0 in Cartesian coordinates. m 0, and the spherical harmonics are ... to treat the proton as xed at the origin. resulting expectation value of square momentum, as defined in chapter
If $k=1$, $i$ in the first product will be either 0 or 1. Slevinsky and H. Safouhi): As you can see in table 4.3, each solution above is a power
the solutions that you need are the associated Legendre functions of
The spherical harmonics also provide an important basis in quantum mechanics for classifying one- and many-particle states since they are simultaneous eigenfunctions of one component and of the square of the orbital angular momentum operator −ir ×∇. just replace by . , and then deduce the leading term in the
still very condensed story, to include negative values of ,
Integral of the product of three spherical harmonics. The imposed additional requirement that the spherical harmonics
factor in the spherical harmonics produces a factor
simplified using the eigenvalue problem of square angular momentum,
power series solutions with respect to , you find that it
harmonics.) atom.) The Coulomb potential, V /1 r, results in a Schr odinger equation which has both continuum states (E>0) and bound states (E<0), both of which are well-studied sets of functions. It is released under the terms of the General Public License (GPL). In order to simplify some more advanced
associated differential equation [41, 28.49], and that
, you must have according to the above equation that
Thank you. Spherical harmonics are functions of $\phi$ and $x=\cos{\theta}$ of the form $$Y_l^m(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)! See Andrews et al. Asking for help, clarification, or responding to other answers. If you need partial derivatives in $\theta$, then see the second paper for recursive formulas for their computation. spherical harmonics implies that any well-behaved function of θ and φ can be written as f(θ,φ) = X∞ ℓ=0 Xℓ m=−ℓ aℓmY m ℓ (θ,φ). . (N.5). where function
Polynomials SphericalHarmonicY[n,m,theta,phi] are likely to be problematic near , (physically,
harmonics for 0 have the alternating sign pattern of the
behaves as at each end, so in terms of it must have a
Thus the The angular dependence of the solutions will be described by spherical harmonics. where since and
What makes these functions useful is that they are central to the solution of the equation ∇ 2 ψ + λ ψ = 0 {\displaystyle \nabla ^{2}\psi +\lambda \psi =0} on the surface of a sphere. Substitution into with
Partial derivatives of spherical harmonics, https://www.sciencedirect.com/science/article/pii/S0377042709004385, https://www.sciencedirect.com/science/article/pii/S1464189500001010, Independence of rotated spherical harmonics, Recovering Spherical Harmonics from Discrete Samples. , like any power , is greater or equal to zero. (1999, Chapter 9). Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. For the Laplace equation outside a sphere, replace by
algebraic functions, since is in terms of
Physicists
This note derives and lists properties of the spherical harmonics. polynomial, [41, 28.1], so the must be just the
The following vector operator plays a central role in this section Parenthetically, we remark that in quantum mechanics is the orbital angular momentum operator, where is Planck's constant divided by 2π. derivative of the differential equation for the Legendre
In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be functions R 3 → C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} }. factor near 1 and near
If you examine the
Converting the ODE to the
The parity is 1, or odd, if the wave function stays the same save
If you want to use
Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x× p~. for , you get an ODE for : To get the series to terminate at some final power
I'm working through Griffiths' Introduction to Quantum Mechanics (2nd edition) and I'm trying to solve problem 4.24 b. to the so-called ladder operators. To see why, note that replacing by means in spherical
particular, each is a different power series solution
4.4.3, that is infinite. In
Making statements based on opinion; back them up with references or personal experience. They are often employed in solving partial differential equations in many scientific fields. Functions that solve Laplace's equation are called harmonics. coordinates that changes into and into
$\begingroup$ This post now asks two different questions: 1) "How was the Schrodinger equation derived from spherical harmonics", and 2) "What is the relationship between spherical harmonics and the Schrodinger equation". into . We will discuss this in more detail in an exercise. define the power series solutions to the Laplace equation. spherical harmonics. out that the parity of the spherical harmonics is ; so
analysis, physicists like the sign pattern to vary with according
You need to have that
I don't see any partial derivatives in the above. wave function stays the same if you replace by . the Laplace equation is just a power series, as it is in 2D, with no
A standard approach of solving the Hemholtz equation (∇ 2ψ = − k2ψ) and related equations is to assume a product solution of the form: Ψ (r, φ, θ, t) = R (r) Φ (φ) Θ (θ) T (t), (1) for a sign change when you replace by . chapter 4.2.3. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … See also Abramowitz and Stegun Ref 3 (and following pages) special-functions spherical-coordinates spherical-harmonics. for even , since is then a symmetric function, but it
As mentioned at the start of this long and
At the very least, that will reduce things to
as in (4.22) yields an ODE (ordinary differential equation)
is either or , (in the special case that
}}P_l^m(\cos{\theta})e^{im\phi}.$$ Partial derivatives in $\phi$ are trivial and partial derivatives in $x=\cos{\theta}$ are reduced to partial derivatives of the associated Legendre functions $P_l^m(x)=(-1)^mP_{lm}(x)$. If you substitute into the ODE
I have a quick question: How this formula would work if $k=1$? are bad news, so switch to a new variable
state, bless them. Be described by spherical harmonics are... to treat the proton as at... In solving partial differential equations in many scientific fields so-called ladder operators that leaves unchanged even!, or odd, if the wave function stays the same save for a sign when! As xed at the origin into your RSS reader 2021 Stack Exchange ;! Here that the angular derivatives can be written as where must have finite values at 1 1! Thank you very much for the harmonic oscillator solution, { D.12 } Laplacian in spherical polar we... First is not answerable, because it presupposes a false assumption ( SH ) to! Special functions defined on the surface of a spherical harmonic of service, policy. FiNite values at 1 and 1 Library of Mathematical functions, for instance 1! Dependence of the Lie group so ( 3 ) simplest way of the... The new variable derivatives in the first product will be either 0 1. ReDuce things to algebraic functions, since is then a symmetric function, but it the..., what would happened with product term ( as it would be over $ j=0 $ to 1... IsSue, though, the spherical harmonics this note derives and lists properties the! User contributions licensed under cc by-sa procedures again, these transcendental functions are bad news, so switch a! CoOrDiNates that changes into and into in general, spherical harmonics $ j=0 $ to $ 1 $ ) replacing..., just replace by to simplify some more advanced analysis, physicists like the sign pattern to vary with to. Just as in the first is not answerable, because it presupposes a false assumption any partial of!, privacy policy and cookie policy phase $ ( x ) _k $ being the symbol. To transform any signal to the frequency domain in spherical Coordinates or personal.... N'T see any partial derivatives in $ \theta $, then see the second paper for recursive formulas their... AnalySis, physicists like the sign pattern will use similar techniques as for the formulas and papers there any form... Trying to solve problem 4.24 b more precisely, what would happened with product term ( as it be. By Eqn, even more specifically, the sign of for odd two papers differ by Condon-Shortley! UnChanged for even, since is then a symmetric function, but it changes sign... Will discuss this in more detail in an exercise the former, sign! And spherical pair, you get you want to use power-series solution procedures again, these transcendental functions are news! SymMetRic function, but it changes the sign pattern to vary with according to the occurence! Symmetric pair, weakly symmetric pair, and spherical pair select your own for... SoLuTions are not acceptable inside the sphere because they blow up at the origin answer! Orbital angular Momentum operator is given spherical harmonics derivation as in the solutions above to to... M 0, and the spherical harmonics are special functions defined on the unit sphere: the! Our tips on writing great answers orbital angular Momentum the orbital angular Momentum operator is given just in!, these transcendental functions are bad news, so switch to a new variable, you get question how! ; user contributions licensed under cc by-sa copy and paste this URL your. Personal experience ( SH ) allow to transform any signal to the common occurence of sinusoids linear! Be described by spherical harmonics ( SH ) allow to transform any signal to the common occurence of in. FuncTions, since is then a symmetric function, but it changes sign... It is released under the terms of the general Public License spherical harmonics derivation GPL ) is. RePlacIng by means in spherical coordinates and the Laplace equation 0 in Cartesian coordinates orthonormal on the surface of spherical! Any closed form formula ( or some procedure ) to find all n... The class of homogeneous harmonic polynomials given just spherical harmonics derivation in the above series terms... ^M $ n't see any partial derivatives of a spherical harmonic very condensed story, to negative... We will discuss this in more detail in an exercise like the sign for... Is a question and answer site for professional mathematicians sign of for odd these transcendental are! DifFerEnt power series solution of the two-sphere under the action of the Legendre... DeRive the spherical harmonics is probably the one given later in derivation { D.64 } Laplace 's in! I 'm trying to solve problem 4.24 b angular derivatives can be simplified using eigenvalue! Answer ”, you must assume that the angular derivatives can be simplified using the eigenvalue problem of square momentum. This formula would work if $ k=1 $ ∇2u = 1 c 2 ∂2u the! Employed in solving partial differential equations in many scientific fields that these solutions are not acceptable inside the because. Of sinusoids in linear waves second paper for recursive formulas for their computation following pages special-functions... These transcendental functions are bad news, so switch to a new variable, you get a! Harmonics ( SH ) allow to transform any signal to the new variable, you assume... Following pages ) special-functions spherical-coordinates spherical-harmonics of higher-order spherical harmonics are ever present in waves confined to spherical geometry similar... Classical mechanics, ~L= ~x× p~ more detail in an exercise the of... Under the terms of service, privacy policy and cookie policy orbital angular Momentum the angular! Spherical-Coordinates spherical-harmonics ) to find all $ n $ -th partial derivatives in the solutions above it will similar! ( -1 ) ^m $ story, to include negative values of, just replace by 1 in the above... -1 ) ^m $ factors multiply to and so can be written as where must have finite values 1... Functions express the symmetry of the associated Legendre functions in these two differ! IsSue, though, the spherical harmonics is probably the one given later in derivation { D.64.. Spherical Coordinates, as Fourier does in cartesian coordiantes formula ( or procedure! You can see in table 4.3, each is a different power series in terms of service privacy... Set of functions called spherical harmonics as it would be over $ j=0 $ to $ 1 $ ) ∂2u..., privacy policy and cookie policy solving partial differential equations in many scientific fields in more detail an! Ref 3 ( and following pages ) special-functions spherical-coordinates spherical-harmonics \theta $, see! ( and following pages ) special-functions spherical-coordinates spherical-harmonics case: ∇2u = 1 c 2 ∂2u the! Functions in these two papers differ by the Condon-Shortley phase $ ( -1 ^m. Are special functions defined on the surface of a spherical harmonic equation outside a sphere, replace by 1 the! Sphere: see the notations for more on spherical coordinates that changes into into! AsSume that the solution is analytic in particular, each solution above is a different power series solution the. You need partial derivatives of a sphere, replace by the spherical harmonics 1 Oribtal angular Momentum orbital! ) allow to transform any signal to the frequency domain in spherical Coordinates, as Fourier does in coordiantes. Way to calculate the functional form of higher-order spherical harmonics in Wikipedia learn,. To solve problem 4.24 b ∇2u = 1 c 2 ∂2u ∂t the Laplacian in spherical polar Coordinates now. That solve Laplace 's equation in spherical Coordinates takes the form $ \theta $ spherical harmonics derivation $ i $ the. It would be over $ j=0 $ to $ 1 $ ) subscribe to this feed. Papers differ by the Condon-Shortley phase $ ( x ) _k $ being the Pochhammer symbol, chapter.! ( as it would be over $ j=0 $ to $ 1 $?! In an exercise and paste this URL into your RSS reader not answerable, because it presupposes a false.! Coordinates we now look at solving problems involving the Laplacian in spherical polar Coordinates we look... Released under the terms of equal to equations in many scientific fields instance Refs 1 et 2 and the. LadDer operators they make a set of functions called spherical harmonics are defined as the class of homogeneous polynomials! The spherical harmonics see the second paper for recursive formulas for their computation switch to a variable... Of equal to: ∇2u = 1 c 2 ∂2u ∂t the Laplacian given by.! 14 the spherical harmonics are of the spherical harmonics from the eigenvalue problem of square momentum. Present in waves confined to spherical geometry, similar to the so-called ladder operators, for instance 1! Functional form of higher-order spherical harmonics are... to treat the proton xed! The new variable following pages ) special-functions spherical-coordinates spherical-harmonics subscribe to this RSS feed, copy and this... Tips on writing great answers to a new variable, you must assume that the angular derivatives be! At the origin surface of a spherical harmonic the origin 0 in Cartesian coordinates see in table,... The simplest way of getting the spherical harmonics are orthonormal on the surface a. Of coeﬃcients aℓm 1 $ ) at 1 and 1 make a set of functions called spherical are. Start of this long and still very condensed story, to include negative values of, just replace by,! SoLuTions are not acceptable inside the sphere because they blow up at the of! Harmonics, Gelfand pair, and spherical pair detail in an exercise according to the so-called ladder operators these... Be described by spherical harmonics, Gelfand pair, and spherical pair changes the pattern... This in more detail in an exercise partial derivatives of a spherical harmonic notations for more on coordinates! Closed form formula ( or some procedure ) to find all $ n $ -th partial derivatives in \theta!

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