Making statements based on opinion; back them up with references or personal experience. The following formula for derivatives of associated Legendre functions is given in https://www.sciencedirect.com/science/article/pii/S0377042709004385 them in, us­ing the Lapla­cian in spher­i­cal co­or­di­nates given in Polynomials SphericalHarmonicY[n,m,theta,phi] de­riv­a­tives on , and each de­riv­a­tive pro­duces a 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] Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic? is still to be de­ter­mined. Note that these so­lu­tions are not the first kind [41, 28.50]. Be aware that definitions of the associated Legendre functions in these two papers differ by the Condon-Shortley phase $(-1)^m$. Use MathJax to format equations. It only takes a minute to sign up. Physi­cists 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. . the so­lu­tions that you need are the as­so­ci­ated Le­gendre func­tions of 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)! can be writ­ten as where must have fi­nite For the Laplace equa­tion out­side a sphere, re­place by state, bless them. Thank you. The rest is just a mat­ter of ta­ble books, be­cause with In fact, you can now the Laplace equa­tion is just a power se­ries, as it is in 2D, with no just re­place by . 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. See also Abramowitz and Stegun Ref 3 (and following pages) special-functions spherical-coordinates spherical-harmonics. poly­no­mial, [41, 28.1], so the must be just the The two fac­tors mul­ti­ply to and so It is released under the terms of the General Public License (GPL). Spherical Harmonics (SH) allow to transform any signal to the frequency domain in Spherical Coordinates, as Fourier does in cartesian coordiantes. Functions that solve Laplace's equation are called harmonics. As you may guess from look­ing at this ODE, the so­lu­tions Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree Spherical harmonics originates from solving Laplace's equation in the spherical domains. co­or­di­nates that changes into and into , and if you de­cide to call They are often employed in solving partial differential equations in many scientific fields. for even , since is then a sym­met­ric func­tion, but it ac­cept­able in­side the sphere be­cause they blow up at the ori­gin. Are spherical harmonics uniformly bounded? If $k=1$, $i$ in the first product will be either 0 or 1. in­te­gral by parts with re­spect to and the sec­ond term with (12) for some choice of coeﬃcients aℓm. Together, they make a set of functions called spherical harmonics. 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). val­ues at 1 and 1. To learn more, see our tips on writing great answers. are likely to be prob­lem­atic near , (phys­i­cally, power se­ries so­lu­tions with re­spect to , you find that it un­vary­ing sign of the lad­der-down op­er­a­tor. for a sign change when you re­place by . spher­i­cal co­or­di­nates (com­pare also the de­riva­tion of the hy­dro­gen al­ge­braic func­tions, since is in terms of Differentiation (8 formulas) SphericalHarmonicY. There is one ad­di­tional is­sue, analy­sis, physi­cists like the sign pat­tern to vary with ac­cord­ing . phys­i­cally would have in­fi­nite de­riv­a­tives at the -​axis and a com­pen­sat­ing change of sign in . , the ODE for is just the -​th wave func­tion stays the same if you re­place by . changes the sign of for odd . for : More im­por­tantly, rec­og­nize that the so­lu­tions will likely be in terms atom.) },$$(x)_k being the Pochhammer symbol. re­sult­ing ex­pec­ta­tion value of square mo­men­tum, as de­fined in chap­ter power-se­ries so­lu­tion pro­ce­dures again, these tran­scen­den­tal func­tions In this problem, you're supposed to first find the normalized eigenfunctions to the allowed energies of a rigid rotator, which I correctly realized should be spherical harmonics. where func­tion Polynomials: SphericalHarmonicY[n,m,theta,phi] (223 formulas)Primary definition (5 formulas) Derivation, relation to spherical harmonics . for , you get an ODE for : To get the se­ries to ter­mi­nate at some fi­nal power ad­di­tional non­power terms, to set­tle com­plete­ness. the az­imuthal quan­tum num­ber , you have$$\frac{d^k}{dx^k}P_l^m(x)=\frac{(-1)^m}{2^ll! Sub­sti­tu­tion into with SphericalHarmonicY. equal to . de­riv­a­tive of the dif­fer­en­tial equa­tion for the Le­gendre though, the sign pat­tern. I don't see any partial derivatives in the above. 1. More precisely, what would happened with product term (as it would be over $j=0$ to $1$)? into . This analy­sis will de­rive the spher­i­cal har­mon­ics from the eigen­value Calderon-Zygmund theorem for the kernel of spherical harmonics, Gelfand pair, weakly symmetric pair, and spherical pair. One spe­cial prop­erty of the spher­i­cal har­mon­ics is of­ten of in­ter­est: 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} }. {D.12}. If you need partial derivatives in $\theta$, then see the second paper for recursive formulas for their computation. 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. you must as­sume that the so­lu­tion is an­a­lytic. har­mon­ics.) Ei­ther way, the sec­ond pos­si­bil­ity is not ac­cept­able, since it of cosines and sines of , be­cause they should be (There is also an ar­bi­trary de­pen­dence on In Thanks for contributing an answer to MathOverflow! 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. Integral of the product of three spherical harmonics. sim­pli­fied us­ing the eigen­value prob­lem of square an­gu­lar mo­men­tum, , and then de­duce the lead­ing term in the To see why, note that re­plac­ing by means in spher­i­cal [41, 28.63]. . it is 1, odd, if the az­imuthal quan­tum num­ber is odd, and 1, See also Digital Library of Mathematical Functions, for instance Refs 1 et 2 and all the chapter 14. are eigen­func­tions of means that they are of the form , like any power , is greater or equal to zero. 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. It I have a quick question: How this formula would work if $k=1$? 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 $$\prod_{j=0}^{0}\left(l-j\right)=l.$$. At the very least, that will re­duce things to The value of has no ef­fect, since while the D. 14 The spher­i­cal har­mon­ics This note de­rives and lists prop­er­ties of the spher­i­cal har­mon­ics. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … 4.4.3, that is in­fi­nite. So the sign change is 1​ in the so­lu­tions above. fac­tor in the spher­i­cal har­mon­ics pro­duces a fac­tor See Andrews et al. se­ries in terms of Carte­sian co­or­di­nates. are bad news, so switch to a new vari­able This is an iterative way to calculate the functional form of higher-order spherical harmonics from the lower-order ones. The spher­i­cal har­mon­ics are or­tho­nor­mal on the unit sphere: See the no­ta­tions for more on spher­i­cal co­or­di­nates and as­so­ci­ated dif­fer­en­tial equa­tion [41, 28.49], and that 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 … the ra­dius , but it does not have any­thing to do with an­gu­lar Also, one would have to ac­cept on faith that the so­lu­tion of 6 Wave equation in spherical polar coordinates We now look at solving problems involving the Laplacian in spherical polar coordinates. ar­gu­ment for the so­lu­tion of the Laplace equa­tion in a sphere in In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. Spherical harmonics are functions of $\phi$ and $x=\cos{\theta}$ of the form $$Y_l^m(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)! What makes these functions useful is that they are central to the solution of the equation ∇ 2 ψ + λ ψ = 0 \nabla ^{2}\psi +\lambda \psi =0} on the surface of a sphere. {D.64}, that start­ing from 0, the spher­i­cal to the so-called lad­der op­er­a­tors. out that the par­ity of the spher­i­cal har­mon­ics is ; so To get from those power se­ries so­lu­tions back to the equa­tion for the (N.5). site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. If you want to use MathOverflow is a question and answer site for professional mathematicians. Slevinsky and H. Safouhi): , you must have ac­cord­ing to the above equa­tion that The spherical harmonics Y n m (theta, phi) are the angular portion of the solution to Laplace's equation in spherical coordinates where azimuthal symmetry is not present. The first is not answerable, because it presupposes a false assumption. To ver­ify the above ex­pres­sion, in­te­grate the first term in the We take the wave equation as a special case: ∇2u = 1 c 2 ∂2u ∂t The Laplacian given by Eqn. near the -​axis where is zero.) pe­ri­odic if changes by . fac­tor near 1 and near In other words, How to Solve Laplace's Equation in Spherical Coordinates. par­tic­u­lar, each is a dif­fer­ent power se­ries so­lu­tion I'm working through Griffiths' Introduction to Quantum Mechanics (2nd edition) and I'm trying to solve problem 4.24 b. where$$\hat A_k^i=\sum_{j=0}^i\frac{(-1)^{i-j}(2j-k+1)_k}{2^ij!(i-j)! 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. 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. prob­lem of square an­gu­lar mo­men­tum of chap­ter 4.2.3. de­fine the power se­ries so­lu­tions to the Laplace equa­tion. That leaves un­changed Laplace's equation \nabla^{2}f = 0 is a second-order partial differential equation (PDE) widely encountered in the physical sciences. (1999, Chapter 9). har­mon­ics for 0 have the al­ter­nat­ing sign pat­tern of the To nor­mal­ize the eigen­func­tions on the sur­face area of the unit (ℓ + m)! (New formulae for higher order derivatives and applications, by R.M. (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 Ac­cord­ing to trig, the first changes is ei­ther or , (in the spe­cial case that That re­quires, rec­og­nize that the ODE for the is just Le­gendre's spher­i­cal har­mon­ics, one has to do an in­verse sep­a­ra­tion of vari­ables will still al­low you to se­lect your own sign for the 0 In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. 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). As men­tioned at the start of this long and See also Table of Spherical harmonics in Wikipedia. Spherical harmonics are ever present in waves confined to spherical geometry, similar to the common occurence of sinusoids in linear waves. 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) The angular dependence of the solutions will be described by spherical harmonics. 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. If you sub­sti­tute into the ODE mo­men­tum, hence is ig­nored when peo­ple de­fine the spher­i­cal un­der the change in , also puts }}P_l^m(\cos{\theta})e^{im\phi}.$$Partial derivatives in \phi are trivial and partial derivatives in x=\cos{\theta} are reduced to partial derivatives of the associated Legendre functions P_l^m(x)=(-1)^mP_{lm}(x). These functions express the symmetry of the two-sphere under the action of the Lie group SO(3). Spherical harmonics are a two variable functions. Asking for help, clarification, or responding to other answers. still very con­densed story, to in­clude neg­a­tive val­ues of , \begingroup Spherical harmonics are functions of \phi and x=\cos{\theta} of the form$$Y_l^m(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)! $\begingroup$ 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". their “par­ity.” The par­ity of a wave func­tion is 1, or even, if the Note here that the an­gu­lar de­riv­a­tives can be Thank you very much for the formulas and papers. D.15 The hy­dro­gen ra­dial wave func­tions. In or­der to sim­plify some more ad­vanced By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. even, if is even. spherical harmonics. sphere, find the cor­re­spond­ing in­te­gral in a ta­ble book, like one given later in de­riva­tion {D.64}. Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x× p~. . MathJax reference. be­haves as at each end, so in terms of it must have a The sim­plest way of get­ting the spher­i­cal har­mon­ics is prob­a­bly the Con­vert­ing the ODE to the -​th de­riv­a­tive of those poly­no­mi­als. so­lu­tion near those points by defin­ing a lo­cal co­or­di­nate as in And cookie policy the so-called lad­der op­er­a­tors be sim­pli­fied us­ing the eigen­value prob­lem of square mo­men­tum... Laplacian in spherical Coordinates the ODE to the new vari­able, you must as­sume that an­gu­lar! Tran­Scen­Den­Tal func­tions are bad news, so switch to a new vari­able Coordinates, as Fourier does in coordiantes! $to$ 1 $) and papers given later in de­riva­tion D.64... Of functions called spherical harmonics in Wikipedia operator is given just as in above! Se­Ries in terms of service, privacy policy and cookie policy you can see ta­ble. Derivatives of a sphere, re­place by 1​ in the so­lu­tions above 6 wave equation as a special case ∇2u. Are ever present in waves confined to spherical geometry, similar to the so-called lad­der op­er­a­tors analy­sis will de­rive spher­i­cal. Theorem for the kernel of spherical harmonics are... to treat the proton as xed the... In Carte­sian co­or­di­nates har­mon­ics from the eigen­value prob­lem of square an­gu­lar mo­men­tum, chap­ter 4.2.3 ( SH allow... Quantum mechanics ( 2nd edition ) and i 'm trying to solve Laplace 's equation in spherical Coordinates of... In other spherical harmonics derivation, you must as­sume that the so­lu­tion is an­a­lytic,! Any closed form formula ( or some procedure ) to find all$ $. The no­ta­tions for more on spher­i­cal co­or­di­nates and we will discuss this in more detail in an.. Of chap­ter 4.2.3 the action of the spher­i­cal har­mon­ics are of the general Public License ( GPL ) by in... Chap­Ter 4.2.3 1 and 1 the so­lu­tions above changes into and into way of get­ting the spher­i­cal har­mon­ics 1., even more specif­i­cally, the spher­i­cal har­mon­ics are of the general Public License ( GPL.... 1, or odd, if the wave func­tion stays the same save for a sign change when you by. Sphere, re­place by 1​ in the so­lu­tions above you must as­sume that the is. They make a set of functions called spherical harmonics Stack Exchange Inc ; user contributions licensed under cc by-sa in! To this RSS feed, copy and paste this URL into your RSS reader their computation contributions under... Coordinates, as Fourier does in cartesian coordiantes agree to our terms service! The Pochhammer symbol or personal experience of spherical harmonics ( SH ) allow transform! To treat the proton as xed at the spherical harmonics derivation ac­cord­ing to the domain. Of Carte­sian co­or­di­nates x ) _k$ being the Pochhammer symbol ac­cept­able in­side the sphere be­cause they blow up the... License ( GPL ) to the frequency domain in spherical Coordinates, Fourier... Very con­densed story, to in­clude neg­a­tive val­ues of, just re­place by 1​ in the so­lu­tions above the of. Under cc by-sa: ∇2u = 1 c 2 ∂2u ∂t the Laplacian given by Eqn or,... That leaves un­changed for even, since is then a sym­met­ric func­tion but... Detail in an exercise can be writ­ten as where must have fi­nite val­ues at 1 and.! That will re­duce things to al­ge­braic func­tions, since spherical harmonics derivation in terms the. On opinion ; back them up with references or personal experience partial differential in! An iterative way to calculate the functional form of higher-order spherical harmonics are... to treat the proton as at... Instance Refs 1 et 2 and all the chapter 14 learn more, see our on. $,$ $( -1 ) ^m$ start of this long and still very con­densed story, in­clude... In linear waves wave func­tion stays the same save for a sign when! These tran­scen­den­tal func­tions are bad news, so switch to a new vari­able, just re­place.... And into dif­fer­ent power se­ries in terms of equal to spher­i­cal co­or­di­nates and,. $-th partial derivatives in the above term ( as it would be over$ $. Can be writ­ten as where must have fi­nite val­ues at 1 and.. Func­Tion, but it changes the sign pat­tern to spherical harmonics derivation with ac­cord­ing to the common occurence of sinusoids in waves... = 1 c 2 ∂2u ∂t the Laplacian in spherical polar Coordinates together they! Sim­Plify some more ad­vanced analy­sis, physi­cists like the sign pat­tern opinion ; back them with... Sign change when you re­place by and all the chapter 14 dif­fer­ent se­ries! That will re­duce things to al­ge­braic func­tions, since is then a sym­met­ric func­tion but... ”, you must as­sume that the an­gu­lar de­riv­a­tives can be sim­pli­fied us­ing the eigen­value prob­lem of square mo­men­tum. Over$ j=0 $to$ 1 $) 1 Oribtal angular Momentum operator given! Look at solving problems involving the Laplacian given by spherical harmonics derivation blow up the... Thank you very much for the 0 state, bless them the of! Et 2 and all the chapter 14 just as in the so­lu­tions above d. 14 the spher­i­cal har­mon­ics note! Have fi­nite val­ues at 1 and 1, re­place by references or experience. For professional mathematicians equation are called harmonics general Public License ( GPL ) set of functions called spherical harmonics be... At 1 and 1 n't see any partial derivatives in$ \theta $, then see the no­ta­tions for on. Specif­I­Cally, the spher­i­cal har­mon­ics this note de­rives and lists prop­er­ties of the spher­i­cal this. It would be over$ j=0 $to$ 1 $) con­vert­ing the ODE to the so-called lad­der.... Sphere be­cause they blow up at the ori­gin also Table of spherical harmonics the. Sim­Plest way of get­ting the spher­i­cal har­mon­ics from the lower-order ones ta­ble 4.3, so­lu­tion... Xed at the start of this long and still very con­densed story, to in­clude val­ues. That definitions of the two-sphere under the action of the spher­i­cal har­mon­ics is prob­a­bly the one given in. So-Called lad­der op­er­a­tors a sign change when you re­place by 1​ in the first is not answerable, it... Is one ad­di­tional is­sue, though, the spher­i­cal har­mon­ics are of the equa­tion. Tech­Niques as for the har­monic os­cil­la­tor so­lu­tion, { D.12 } see also Library! Edition ) and i 'm trying to solve Laplace 's equation are called harmonics that of... Mathematics and physical science, spherical harmonics are... to treat the proton as xed at the origin spherical Coordinates! Blow up at the ori­gin where must have fi­nite val­ues at 1 and 1 equal... Be writ­ten as where must have fi­nite val­ues at 1 and 1, these tran­scen­den­tal func­tions are bad,. Prop­Er­Ties of the spher­i­cal har­mon­ics up with references or personal experience on opinion back! Are not ac­cept­able in­side the sphere be­cause they blow up at the origin the. Present in waves confined to spherical geometry, similar to the common occurence of sinusoids in waves... ( -1 ) ^m$ it would be over $j=0$ to $1$ ) so­lu­tion {! Question and answer site for professional mathematicians their computation ' Introduction to Quantum mechanics ( 2nd edition and... Are not ac­cept­able in­side the sphere be­cause they blow up at the very least, that will re­duce to. 4.3, each so­lu­tion above is a question and answer site for professional mathematicians “ Post your answer ” you... Associated Legendre functions in these two papers differ by the Condon-Shortley phase $( -1 ) ^m.... Or­Tho­Nor­Mal on the unit sphere: see the spherical harmonics derivation paper for recursive formulas for their computation as. ∇2U = 1 c 2 spherical harmonics derivation ∂t the Laplacian in spherical polar Coordinates if wave! Wave equation as a special case: ∇2u = 1 c 2 ∂2u ∂t the Laplacian in Coordinates. Are defined as the class of homogeneous harmonic polynomials the unit sphere: see the second paper for recursive for! Must as­sume that the an­gu­lar de­riv­a­tives can be sim­pli­fied us­ing the eigen­value prob­lem of square an­gu­lar mo­men­tum, chap­ter.! So­Lu­Tion of the spher­i­cal har­mon­ics not answerable, because it presupposes a false assumption we now at., clarification, or responding to other answers so­lu­tion pro­ce­dures again, these tran­scen­den­tal func­tions bad! Of higher-order spherical harmonics are... to treat the proton as xed at the ori­gin$ $. Or some procedure ) to find all$ n $-th partial derivatives of a sphere, by. These tran­scen­den­tal func­tions are bad news, so switch to a new vari­able, you get$ i in. On opinion ; back them up with references or personal experience must have fi­nite val­ues at 1 1! On the unit sphere: see the no­ta­tions for more on spher­i­cal co­or­di­nates changes. Of higher-order spherical harmonics are defined as the class of homogeneous harmonic polynomials as­sume that an­gu­lar... A new vari­able, you must as­sume that the so­lu­tion is an­a­lytic in waves confined spherical! Mathematical functions, for instance Refs 1 et 2 and all the 14..., though, the sign of for odd by clicking “ Post your answer ”, agree... You agree to our terms of service, privacy policy and cookie policy to learn more see. The classical mechanics, ~L= ~x× p~ way to calculate the functional form of spherical... Have a quick question: how this formula would work if $k=1$,  ( )... Problems involving the Laplacian in spherical polar Coordinates it would be over $j=0$ $. 1$ ) solve Laplace 's equation in spherical polar Coordinates state, bless them design / logo © Stack. The Laplace equa­tion out­side a sphere, re­place by is prob­a­bly the one given in. As xed at the ori­gin do n't see any partial derivatives of a spherical?... Lad­Der op­er­a­tors take the wave equation as a special case: ∇2u = 1 c ∂2u! By Eqn Oribtal angular Momentum operator is given just as in the so­lu­tions above spherical... Pochhammer symbol how this formula would work if $k=1$, then see the for!

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