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Table Of Spherical Harmonics
This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree \ell = 10. Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in ''x'', ''y'', ''z'', and ''r''. For purposes of this table, it is useful to express the usual spherical to Cartesian transformations that relate these Cartesian components to \theta and \varphi as \begin \cos(\theta) & = z/r\\ e^ \cdot \sin(\theta) & = (x \pm iy)/r \end Complex spherical harmonics For ''ℓ'' = 0, …, 5, see. ''ℓ'' = 0 Y_^(\theta,\varphi)=\sqrt ''ℓ'' = 1 \begin Y_^(\theta,\varphi) &= & & \sqrt\cdot e^\cdot\sin\theta & &= & &\sqrt \cdot \\ Y_^(\theta,\varphi) &= & & \sqrt\cdot \cos\theta & &= & &\sqrt \cdot \\ Y_^(\theta,\varphi) &= &-& \sqrt\cdot e^\cdot \sin\theta & &= &-&\sqrt \cdot \end ''ℓ'' = 2 \begin Y_^(\theta,\varphi)&=& &\sqrt\cdot e^\cdot\sin^\theta\quad && ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Spherical harmonics originate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Common Coordinate Transformations
This is a list of some of the most commonly used coordinate transformations. 2-dimensional Let (''x'', ''y'') be the standard Cartesian coordinates, and (''r'', ''θ'') the standard polar coordinates. To Cartesian coordinates From polar coordinates :\begin x &= r\cos\theta \\ y &= r\sin\theta \\ pt \frac &= \begin \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end \\ pt \text = \det &= r \end From log-polar coordinates :\begin x &= e^\rho\cos\theta, \\ y &= e^\rho\sin\theta. \end By using complex numbers (x, y) = x + iy', the transformation can be written as : x + iy = e^ That is, it is given by the complex exponential function. From bipolar coordinates :\begin x &= a \frac \\ y &= a \frac \end From 2-center bipolar coordinates :\begin x &= \frac\left(r_1^2 - r_2^2\right) \\ y &= \pm \frac\sqrt \end From Cesàro equation :\begin x &= \int \cos \left int \kappa(s) \,ds\rightds \\ y &= \int \sin \left int \kappa(s) \,ds\rightd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Spherical Harmonics Figure Table Complex 2D
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * ''Complex'' (album), by Montaigne, 2019, and its title track * ''Complex'' (EP), by Rifle Sport, 1985 * "Complex" (song), by Gary Numan, 1979 * Complex Networks, publisher of magazine ''Complex'', now online Biology * Protein–ligand complex, a complex of a protein bound with a ligand * Exosome complex, a multi-protein intracellular complex * Protein complex, a group of two or more associated polypeptide chains * Sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Spherical Harmonics Figure Table Complex Polar Plot
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * ''Complex'' (album), by Montaigne, 2019, and its title track * ''Complex'' (EP), by Rifle Sport, 1985 * "Complex" (song), by Gary Numan, 1979 * Complex Networks, publisher of magazine ''Complex'', now online Biology * Protein–ligand complex, a complex of a protein bound with a ligand * Exosome complex, a multi-protein intracellular complex * Protein complex, a group of two or more associated polypeptide chains * Sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Spherical Harmonics Figure Table Complex Radial Magnitude
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * ''Complex'' (album), by Montaigne, 2019, and its title track * ''Complex'' (EP), by Rifle Sport, 1985 * "Complex" (song), by Gary Numan, 1979 * Complex Networks, publisher of magazine ''Complex'', now online Biology * Protein–ligand complex, a complex of a protein bound with a ligand * Exosome complex, a multi-protein intracellular complex * Protein complex, a group of two or more associated polypeptide chains * Sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Spherical Harmonics Figure Table Complex 2D
Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) * ''Real'' (Belinda Carlisle album) (1993) * ''Real'' (Gorgon City EP) (2013) * ''Real'' (IU EP) (2010) * ''Real'' (Ivy Queen album) (2004) * ''Real'' (Mika Nakashima album) (2013) * ''Real'' (Ednita Nazario album) (2007) * ''Real'' (Jodie Resther album), a 2000 album by Jodie Resther * ''Real'' (Michael Sweet album) (1995) * ''Real'' (The Word Alive album) (2014) * ''Real'', a 2002 album by Israel Houghton recording as Israel & New Breed Songs * "Real" (Goo Goo Dolls song) (2008) * "Real" (Gorgon City song) (2013) * "Real" (Plumb song) (2004) * "Real" (Vivid song) (2012) * "Real" (James Wesley song) (2010) * "Real", a song by Kendrick Lamar from ''Good Kid, M.A.A.D City'' * "Real", a song by NF from ''Therapy Session'' * "R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Spherical Harmonics Figure Table Complex Polar Plot
Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) * ''Real'' (Belinda Carlisle album) (1993) * ''Real'' (Gorgon City EP) (2013) * ''Real'' (IU EP) (2010) * ''Real'' (Ivy Queen album) (2004) * ''Real'' (Mika Nakashima album) (2013) * ''Real'' (Ednita Nazario album) (2007) * ''Real'' (Jodie Resther album), a 2000 album by Jodie Resther * ''Real'' (Michael Sweet album) (1995) * ''Real'' (The Word Alive album) (2014) * ''Real'', a 2002 album by Israel Houghton recording as Israel & New Breed Songs * "Real" (Goo Goo Dolls song) (2008) * "Real" (Gorgon City song) (2013) * "Real" (Plumb song) (2004) * "Real" (Vivid song) (2012) * "Real" (James Wesley song) (2010) * "Real", a song by Kendrick Lamar from ''Good Kid, M.A.A.D City'' * "Real", a song by NF from ''Therapy Session'' * "R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Spherical Harmonics Figure Table Complex Radial Magnitude
Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) * ''Real'' (Belinda Carlisle album) (1993) * ''Real'' (Gorgon City EP) (2013) * ''Real'' (IU EP) (2010) * ''Real'' (Ivy Queen album) (2004) * ''Real'' (Mika Nakashima album) (2013) * ''Real'' (Ednita Nazario album) (2007) * ''Real'' (Jodie Resther album), a 2000 album by Jodie Resther * ''Real'' (Michael Sweet album) (1995) * ''Real'' (The Word Alive album) (2014) * ''Real'', a 2002 album by Israel Houghton recording as Israel & New Breed Songs * "Real" (Goo Goo Dolls song) (2008) * "Real" (Gorgon City song) (2013) * "Real" (Plumb song) (2004) * "Real" (Vivid song) (2012) * "Real" (James Wesley song) (2010) * "Real", a song by Kendrick Lamar from ''Good Kid, M.A.A.D City'' * "Real", a song by NF from ''Therapy Session'' * "R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Spherical harmonics originate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign. History Eric W. Weisstein, the creator of the site, was a physics and astronomy student who got into the habit of writing notes on his mathematical readings. In 1995 he put his notes online and called it "Eric's Treasure Trove of Mathematics." It contained hundreds of pages/articles, covering a wide range of mathematical topics. The site became popular as an extensive single resource on mathematics on the web. Weisstein continuously improved the notes and accepted corrections and comments from online readers. In 1998, he made a contract with CRC Press and the contents of the site were published in print and CD-ROM form, titled "CRC Concise Encyclopedia of Mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serbian Astronomical Journal
The ''Serbian Astronomical Journal'' is a biannual peer-reviewed scientific journal covering astronomy. The journal is the successor of the ''Bulletin Astronomique de Belgrade'' (1992–1998), which was formed by a merger of the ''Bulletin de l'Observatoire Astronomique de Belgrade'' (1936–1991) and the ''Publications of the Department of Astronomy'' (1969–1990). It has been published under the present title since 1998. It is published by the Astronomical Observatory Belgrade and the Department of Astronomy (Faculty of Mathematics, University of Belgrade). It publishes invited reviews, original scientific papers, preliminary reports, and professional papers over the entire range of astronomy, astrophysics, astrobiology, and related fields. Abstracting and indexing The ''Serbian Astronomical Journal'' is abstracted and indexed by Astrophysics Data System, Chemical Abstracts, Referativni Zhurnal, EBSCO databases, Scopus and Thomson Reuters products [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
