analytic function

Examples of analytic function in the following topics:

• Applications of Taylor Series

  • Taylor series expansion can help approximating values of functions and evaluating definite integrals.
  • An analytic function is uniquely extended to a holomorphic function on an open disk in the complex plane.
  • The (truncated) series can be used to compute function values numerically.
  • This is particularly useful in evaluating special mathematical functions (such as Bessel function).
  • As more terms are added to the Taylor polynomial, it approaches the correct function.

• Further Transcendental Functions

  • A transcendental function is a function that is not algebraic.
  • Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.
  • Formally, an analytic function $ƒ(z)$ of the real or complex variables $z_1, \cdots ,z_n$ is transcendental if $z_1, \cdots ,z_n$, $ƒ(z)$ are algebraically independent, i.e., if $ƒ$ is transcendental over the field $C(z_1, \cdots ,z_n)$.
  • A transcendental function is a function that "transcends" algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, power, and root extraction.
  • Bottom panel: Graph of sine function versus angle.

• Taylor and Maclaurin Series

  • Taylor series represents a function as an infinite sum of terms calculated from the values of the function's derivatives at a single point.
  • A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
  • Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.
  • The Taylor series of a function is the limit of that function's Taylor polynomials, provided that the limit exists.
  • A function that is equal to its Taylor series in an open interval (or a disc in the complex plane) is known as an analytic function.

• Series Solutions

  • Suppose further that $\frac{a_1}{a_2}$ and $\frac{a_1}{a_2}$ are analytic functions.
  • The exponential function (in blue), and the sum of the first $n+1$ terms of its Maclaurin power series (in red).

• Taylor Polynomials

  • A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of the function's derivatives.
  • Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
  • Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.
  • Let's assume that the integration of a function ($f(x)$) cannot be performed analytically.
  • The exponential function (in blue) and the sum of the first 9 terms of its Taylor series at 0 (in red).

• Integration Using Tables and Computers

  • While differentiation has easy rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful.
  • These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century.
  • Here are a few examples of integrals in these tables for logarithmic functions:
  • These programs know how to perform almost any integral that can be done analytically or in terms of standard mathematical functions.

• Differentials

  • Differentials are the principal part of the change in a function $y = f(x)$ with respect to changes in the independent variable.
  • where $f'(x)$ is the derivative of $f$ with respect to $x$, and $dx$ is an additional real variable (so that $dy$ is a function of $x$ and $dx$).
  • The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or a particular analytical significance if the differential is regarded as a linear approximation to the increment of a function.
  • Higher-order differentials of a function $y = f(x)$ of a single variable $x$ can be defined as follows:
  • Use implicit differentiation to find the derivatives of functions that are not explicitly functions of $x$

• Nonhomogeneous Linear Equations

  • where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions.
  • In simple cases, for example, where the coefficients $A_1(t)$ and $A_2(t)$ are constants, the equation can be analytically solved.
  • Identify when a second-order linear differential equation can be solved analytically

• Logistic Equations and Population Grown

  • The logistic function is the solution of the following simple first-order non-linear differential equation:
  • More quantitatively, as can be seen from the analytical solution, the logistic curve shows early exponential growth for negative $t$, which slows to linear growth of slope $\frac{1}{4}$ near $t = 0$, then approaches $y = 1$ with an exponentially decaying gap.
  • This is represented by the ceiling past which the function ceases to grow.
  • Describe shape of the logistic function and its use for modeling population growth

• Direction Fields and Euler's Method

  • They can be achieved without solving the differential equation analytically, and serve as a useful way to visualize the solutions.
  • It can be viewed as a creative way to plot a real-valued function of two real variables as a planar picture.