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Heun’s functions anddifferential geometry inMaple15Plamen FizievTheGoal: ToOpen thePadlocksofNature!Department of Theoretical PhysicsUniversity of SofiaandBLTF, JINR, DubnaTalk at XIV Workshop on Computer AlgebraDubna, June 03, 2011

The main question:Where we can find the KEY?TheToolA GOOD NEWSAfter the April 15, 2011we haveMaple 15

Accordint to Maplesoft:http://maplesoft.com/products/maple/new_features/Maple 15 now computes symbolicsolutions to 97% of the 1390 linearand non-linear ODEsMaple also solves these ODEs almost 10times faster than Mathematica.from the famous text:Differentialgleichungen by Kamke.Mathematica ® 8 only handles 79%.or alltogheder ( ) : (a simple Maple calculation)‣ 97% + 79 %;‣ = 176 % ( !!! really a fantastic result !!!)

Heun’s DifferentialEquation:A KEYforHugeamountofPhysicalProblemsfoundbyBorn in Weisbaden April 3, 1859Died in Karsruhe January 10, 1929Zur Theorie der Riemann'schenFunctionen zweiter Ordnungmit Vier Verzweigungs-punktenMath. Ann. 31 (1889) 161-179

The General Heun Equation:

ConfluentHeunEquation:Mathieu functions, spheroidal wavefunctions, and Coulomb spheroidalFunctions are special cases.The UNIQUEFrobenius solutionaround z = 0 :Recurrencerelation:The connectionproblem is stillUNSOLVED !

Bi-Confluent Heun Equation:Exactsolutionsfor φ4Double-Confluent Heun Equation:Theree-Confluent Heun Equation:

24 Mobius transformations z -> f(z) of the independent variable z.These forms of f(z) are:Examples with

Some Exactly Solublein terms of Heun’s functionsphysical problems:1. Hidrogen Molecule2. Wasserstoffmoleculeon3. Two-centre problem in QM (Helium).4. Anharmonic Oscillators in QM and QFT5. Stark Effect6. Repulsion and Attraction of Quantum Levels,7. 3D Hydrodinamical Waves in non-isotermal Atmosphere8. Quantum Diffusion of Kinks9. Cristalline Materials10. In celestial Mechanics: Moon’s motion11. Cologero-Moser-Sutherland System12. Bethe ansatz systems…At present – more than 200 scientific problems !

Heun’s problems in gravity: perturbations of1.Schwarzshild metric: PPF, CQG,2006, J Phys C, 20072. Kerr metric (for s = 0, 1/2, 1,3/2,2) PPF, gr-qc/0902.12773. Reisner-Nortstrom metric (for s = 0, 1/2, 1, 3/2, 2).4. Kerr-Newman metric (for s = 0, 1/2, 1, 3/2, 2).5. De Sitter metric (for s = 0, 1/2, 1, 3/2, 2).6. Reisner-Nortstrom-de Sitter metric (for s = 0, 1/2, 1, 3/2, 2).7. Interior perturbations of all above solutions of EE.- for Schwarzschild: PPF gr-qc/0603003.8. QNM of nonrotating and rotating stars and other compactobjects: naked singularities, superspinars, gravastars,boson stars, soliton stars, quark stars, fuzz-balls, dark stars…9. All D-type metrics - Batic D, Schmid H, 2007 JMP 4810. Relativistic jets: PPF, Staicova, astro-ph:HE/0902.2408astro-ph:HE/0902.241111. Continuous spectrum in TME for s =1/2, 1Borissov, PPF, gr-qc/0902.3617

An essential GENERALIZATION:S. Yu. Slavyanov – A Theorem for all Painleve class of classical equations !Note: All Painleve equations are Euler-Lagrange equations: Slavyanov 1966Hamilton structure of the Painleve equations : Malmquist, 1922

P.F. , CQG, 2006 (Schwarzschild )Denitsa Staicova, P.P.F. , Astrophys Space Sci, 2011 (Kerr)

Examples of Relativistic Jets 1PPF, D. Staicova, astro-ph:HE/0902.2408, BAJ 2010

Discovered by NASA'sSpitzer Space Telescope``tornado-like``object Herbig-Haro 49/50,created fromthe shockwaves of powerfulprotostellar jet hittingthe circum-stellarmedium.PPF, D. Staicova,astro-ph:HE/0902.2411,BAJ 2010

Confluent Heun’s Functions ???Cats eye

Some Maple HeunC problems:HeunC((I)*omega,-(I)*omega+1., (6*I)*omega+1., -((-I+1.*omega))*omega, -20.*omega^2-(1.*I)*omega+.5+omega,z))1. For large |z| = 1..100 :2. HeunCPrime=fdif(HeunC), but PPF JPA 20113. Some values of z are problematic(for example) :HeunC(13.7629973824+.199844789*I, -12.7629973824-.199844789*I, -1.0+0.*I, 108.45307688652939865438+2.9503080968932803136*I,-107.95307688652939865438-2.9503080968932803136*I, 110.988405457376-1.5970801306700*I)Digits:=10;Digits:=32;Digits:=64;Conclusion:We need aNEW CODE !based on new ideas(tested already)-3.216621105*10^(-11)+9.335196121*10^(-12)*I-2.52269564229422256*10^(-12)+5.87236956206153258*10^(-12)*I-1.72317085591748299*10^(-12)+4.00958782709241923*10^(-12)*IHeunC(-0.1e-1+1.*I, 1.01-1.*I, .94+6.*I, -1.0099+.98*I, -18.4880-1.39*I, 90.03) =.360445353243995HeunC(-0.1e-1+1.*I, 1.01-1.*I, .94+6.*I, -1.0099+.98*I, -18.4880-1.39*I, 90.04) = Float(infinity)

Another problem:To find the roots of system oftranscendental equations, written interms of Heun’s functionsArXiv: 1005.5375