Eulerin summia

6533b81ffe1ef96bd1278962

RESEARCH PRODUCT

Eulerin summia

subject

sarjateoria Riemannin zeeta-funktio Bernoullin luvut Fourier-sarjat funktiot

description

Tämän tutkielman tarkoituksena on tarkastella menetelmiä joilla voidaan laskea niin kutsuttuja Eulerin summia. Eulerin summia ovat Riemannin zeeta-funktion arvoja parillisissa ja positiivisissa kokonaislukupisteissä. Vaikka kyseessä on ääretön joukko äärettömiä summia, niin Eulerin summien laskemiseksi on mahdollista johtaa eksplisiittinen kaava. Tämä kaava johdetaan tutkielmassa kahdella eri tavalla: Bernoullin lukuja ja toisaalta Fourier-analyysin tuloksia hyödyntäen. Lisäksi tutkielmassa tarkastellaan muutamia menetelmiä, joilla voidaan laskea yksittäisiä Eulerin summia. Huomionarvoinen maininta on, että vaikka mielivaltainen äärellinen Eulerin osasumma on rationaalinen, niin kaikki Eulerin summat ovat silti transkendenttisiä.

year	journal	country	edition	language
2014-01-01

http://urn.fi/URN:NBN:fi:jyu-201406172059

PEPhysical Sciences and Engineering
- PE2Fundamental Constituents of Matter
  - PE2_1Theory of fundamental interactions
  - PE2_2 Phenomenology of fundamental interactions
  - PE2_3Experimental particle physics with accelerators
  - PE2_4Experimental particle physics without accelerators
  - PE2_6Nuclear, hadron and heavy ion physics
  - PE2_8Gas and plasma physics
  - PE2_5Classical and quantum physics of gravitational interactions
  - PE2_7Nuclear and particle astrophysics
  - PE2_9 Electromagnetism
  - PE2_10Atomic, molecular physics
  - PE2_11Ultra-cold atoms and molecules
  - PE2_18Equilibrium and non-equilibrium statistical mechanics: steady states and dynamics
  - PE2_14Lasers, ultra-short lasers and laser physics
  - PE2_17Metrology and measurement
  - PE2_16Non-linear physics
  - PE2_12Optics, non-linear optics and nano-optics
  - PE2_13Quantum optics and quantum information
  - PE2_15Thermodynamics
- PE1Mathematics
  - PE1_1Logic and foundations
  - PE1_6Geometry and global analysis
  - PE1_8Analysis
  - PE1_7Topology
  - PE1_9Operator algebras and functional analysis
  - PE1_10ODE and dynamical systems
  - PE1_12Mathematical physics
  - PE1_11Theoretical aspects of partial differential equations
  - PE1_13Probability
  - PE1_14Mathematical statistics
  - PE1_15Generic statistical methodology and modelling
  - PE1_4 Algebraic and complex geometry
  - PE1_19Scientific computing and data processing
  - PE1_2Algebra
  - PE1_20Control theory, optimisation and operational research
  - PE1_21Application of mathematics in sciences
  - PE1_22Application of mathematics in industry and society
  - PE1_5Lie groups, Lie algebras
  - PE1_18Numerical analysis
  - PE1_16Discrete mathematics and combinatorics
  - PE1_17Mathematical aspects of computer science
  - PE1_3Number theory
- PE3Condensed Matter Physics
  - PE3_1Structure of solids, material growth and characterisation
  - PE3_2Mechanical and acoustical properties of condensed matter, lattice dynamics
  - PE3_3Transport properties of condensed matter
  - PE3_4Electronic properties of materials, surfaces, interfaces, nanostructures
  - PE3_6Macroscopic quantum phenomena, e.g. superconductivity, superfluidity, quantum Hall effect
  - PE3_5Physical properties of semiconductors and insulators
  - PE3_8Magnetism and strongly correlated systems
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  - PE3_13Structure and dynamics of disordered systems, e.g. soft matter (gels, colloids, liquid crystals), granular matter, liquids, glasses, defects
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- PE6Computer Science and Informatics
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  - PE6_1Computer architecture, embedded systems, operating systems
  - PE6_2Distributed systems, parallel computing, sensor networks, cyber-physical systems
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- PE7Systems and Communication Engineering
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  - PE7_9Man-machine interfaces
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  - PE7_11Components and systems for applications (in e.g. medicine, biology, environment)
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  - PE10_14Earth observations from space/remote sensing
  - PE10_11Geochemistry, cosmochemistry, crystal chemistry, isotope geochemistry, thermodynamics
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  - PE10_12Sedimentology, soil science, palaeontology, earth evolution
- PE11Materials Engineering
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  - PE11_7Engineering of metal oxides
  - PE11_8Engineering of alternative established or emergent materials
  - PE11_9Nanomaterials engineering, e.g. nanoparticles, nanoporous materials, 1D & 2D nanomaterials
  - PE11_10Soft materials engineering, e.g. gels, foams, colloids
  - PE11_11Porous materials engineering, e.g. covalent-organic, metal-organic, porous aromatic frameworks
  - PE11_12Semi-conducting and magnetic materials engineering
  - PE11_13Metamaterials engineering
  - PE11_14Computational methods for materials engineering
LSLife Sciences
- LS1Molecules of Life: Biological Mechanisms, Structures and Functions
  - LS1_2Biochemistry
  - LS1_1Macromolecular complexes including interactions involving nucleic acids, proteins, lipids and carbohydrates
  - LS1_3DNA and RNA biology
  - LS1_4Protein biology
  - LS1_5Lipid biology
  - LS1_6Glycobiology
  - LS1_11Chemical biology
  - LS1_12Protein design
  - LS1_13Early translational research and drug design
  - LS1_14Innovative methods and modelling in molecular, structural and synthetic biology
  - LS1_10Synthetic biology
  - LS1_8Structural biology
  - LS1_9Molecular mechanisms of signalling processes
  - LS1_7Molecular biophysics, biomechanics, bioenergetics
- LS2Integrative Biology: from Genes and Genomes to Systems
  - LS2_1Genetics
  - LS2_3Epigenetics
  - LS2_2Gene editing
  - LS2_4Gene regulation
  - LS2_5Genomics
  - LS2_6Metagenomics
  - LS2_7Transcriptomics
  - LS2_8Proteomics
  - LS2_9Metabolomics
  - LS2_10Glycomics/Lipidomics
  - LS2_11Bioinformatics and computational biology
  - LS2_12Biostatistics
  - LS2_14Genetic diseases
  - LS2_16Innovative methods and modelling in integrative biology
  - LS2_13Systems biology
  - LS2_15Integrative biology for personalised medicine
- LS4Physiology in Health, Disease and Ageing
  - LS4_1Organ and tissue physiology and pathophysiology
  - LS4_4Endocrinology
  - LS4_3Physiology of ageing
  - LS4_2Comparative physiology
  - LS4_5Non-hormonal mechanisms of inter-organ and tissue communication
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  - LS4_13Other non-communicable diseases (except disorders of the nervous system and immunity-related diseases)
- LS3Cell Biology, Development, Stem Cells and Regeneration
  - LS3_4Cell junctions, cell adhesion, the extracellular matrix, cell communication
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  - LS3_15Development of cell-based therapeutic approaches for tissue regeneration
  - LS3_17Theoretical modelling in cellular, developmental and regenerative biology
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  - LS3_1Cell cycle, cell division and growth
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- LS5Neuroscience and Disorders of the Nervous System
  - LS5_1Neuronal cells
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  - LS5_4Neural stem cells
  - LS5_2Glial cells and neuronal-glial communication
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  - LS8_12Microbial ecology and evolution
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  - LS8_15Theoretical developments and modelling in environmental biology, ecology, and evolution
- LS9Biotechnology and Biosystems Engineering
  - LS9_3Bioengineering of cells, tissues, organs and organisms
  - LS9_2Applied genetics, gene editing and transgenic organisms
  - LS9_4Microbial biotechnology and bioengineering
  - LS9_5Food biotechnology and bioengineering
  - LS9_6Marine biotechnology and bioengineering
  - LS9_7Environmental biotechnology and bioengineering
  - LS9_8Applied plant sciences, plant breeding, agroecology and soil biology
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  - LS9_9Plant pathology and pest resistance
  - LS9_11Biomass production and utilisation, biofuels
  - LS9_12Ecotoxicology, biohazards and biosafety
  - LS9_1Bioengineering for synthetic and chemical biology
SHSocial Sciences and Humanities
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