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Fundamentals of Computational and Analytical

Calculations

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A non profit research and teaching institute for theoretical, and computational, Physics and Astronomy.

  • Research�
  • Educational Resources �
  • Teaching

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This talk should have probably listed before the Aesthetic of Mathematics (originally behind the coding series), whereas next week’s talk is still an appropriate time.

Overlapping content:

�

  • We define the functions used in a bit more detail and background�
  • Scientific simulations are constructed using calculations

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Philosophy of calculations

Transformation of numbers

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Basic Arithmetics

  • Addition�
    • Subtraction�
    • Multiplication�
    • Division

​

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Combinations of Arithmetic Calculations

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An infinite series of calculations

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Calculus

  • Derivatives�
  • Integrals
  • The study of continuous change���
  • The study of infinitesimal margins

Quora

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Calculus

  • Derivatives�
  • Integrals

study.com

Calcworkshop

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Formal definition of analytic functions

  • f(x) is continuous at f(a) for any a�
  • convergent�
  • f(x) is infinitely differentiable

​

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Holomorphic Function

  • Complex variable (imaginary numbers a.k.a the dimension of numbers that are equal to √-1 (See the aesthetic of mathematics lecture for more details)�
  • Neighbourhood (A set of values that includes itself)�
  • Domain (The minimum and maximum of the set of numbers)�
  • Coordinate space (a space with specified locations for different values)

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Analytic Complex Functions

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  • Harmonic Functions

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  • Cauchy Riemann

https://www.math.nyu.edu/~cerfon

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Examples of analytic functions

Elementary Functions �

  • Polynomials�
  • Exponential functions�
  • Logarithms�
  • Trigonometric functions�

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Special Functions�

  • Hypergeometric functions�
  • Bessel functions�
  • Gamma functions

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General Mathematical Series

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Polynomials

Polynomial Shape Builder

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Exponential functions and Logarithms

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Trigonometry

sparknotes.com

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Hyperbolic Functions

Hyperbola

analogue

Lumen Learning

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Hypergeometric functions

“a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation” (wikipedia)

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Bessel functions

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Gamma functions

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Tabulations and computations

Discussions of how it works?

​

How were the analytic functions actually tabulated??

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  • Analytical formulas can typically be solved on pen and paper���
  • Non-analytical formulas are solved using computational calculations using algorithms

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Calculus

  • Derivatives�
  • Integrals

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Examples of summations

whitman.edu/mathematics/

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Calculus

  • Derivatives�
  • Integrals

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Solving Differential Equations

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Numerical Methods

  • Implicit
  • Explicit

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List of methods to solve differential equations

Euler’s method

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Domain Transformations

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Fourier Transform

Heavy applications of fourier transforms

Step by step look

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Exercise 1

Evaluating the integral of these functions