Jean-Baptiste Joseph Fourier

Transcription

1 Center of Atmospheric Sciences, UNAM September 30, 2016

2 Jean-Baptiste Joseph Jean-Baptiste Joseph ( ) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of series and their applications to problems of heat transfer and vibrations. Son of tailor and orphaned at age 9. Promoted the French revolution. Imprisoned in He claimed: any function of a variable, whether continuous or discontinuous, can be expanded in a series of sines of multiples of the variable.

3 Applications of (start at 3:54)

4 series Consider a function f (x) that is periodic with period. f (x + ) = f (x) the period of the function to 2π, define t = 2π x, so that: f (t + 2π) = f (t) (1) The series are defined as: f (t) = 1 2 a 0 + a k cos(kt) + b k sin(kt) (2) k=1

5 series Consider a function f (x) that is periodic with period. f (x + ) = f (x) the period of the function to 2π, define t = 2π x, so that: f (t + 2π) = f (t) (1) The series are defined as: f (t) = 1 2 a 0 + a k cos(kt) + b k sin(kt) (2) k=1 a 0 = 1 π a k = 1 π b k = 1 π π π π π π π f (x)dx f (x) cos(kt)dt f (x) sin(kt)dt (3) Does this work for every function f? (out of scope) Functions for which first and second order derivatives exist almost everywhere, are finite and have at most a finite number of discontinuities and zero crossings in the interval ( π, π)

6 series Consider a function f (x) that is periodic with period 2. f (x + 2) = f (x) the period of the function to 2π, define t = 2π x = π x, so 2 that: f (t + 2π) = f (t) (4) The series (in this case) are defined as: f (x) = 1 2 a 0 + a 0 = 1 a k = 1 b k = 1 a k cos k=1 f (x)dx ( f (x)cos ( kx f (x)sin ( ) + b k sin ) dx ) dx ( ) (5)

7 Summary of numerical integration Numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral Reasons for numerical integration: 1 The integrand f (x) may be only known in certain points. 2 It may be difficult or impossible to find the analytical antiderivative. 3 It may be easier to compute a numerical approximation than to compute the antiderivative.

8 Summary of numerical integration Combine evaluation of the integrand (f (x)) to get an approximation Evaluate f (x) at certain points and with certain weigths. The number of points and weights depend on the specific method and accuracy required.

9 Summary of numerical integration Combine evaluation of the integrand (f (x)) to get an approximation Evaluate f (x) at certain points and with certain weigths. The number of points and weights depend on the specific method and accuracy required. Some examples are: Midpoint rule: b a+b f (x)dx (b a)f ( ) a 2 Trapezoidal rule: ( ) b f (x)dx (b a) f (a)+f (b) a 2 Composite rule: ( b b a f (x) dx f (a) + n 1 a n 2 k=1 Newton-Cotes Simpson s rule Gaussian quadrature ( f ( a + k b a n )) ) + f (b) 2

10 program series Convince me, using Python, that series are real! (approximate f (x) = x 2 with series) f (x) = 1 a k=1 a k cos ( ) + bk sin ( )

11 program series Convince me, using Python, that series are real! (approximate f (x) = x 2 with series) f (x) = 1 a k=1 a k cos ( ) + bk sin ( ) Step 1 a 0 = 1 f (x)dx (6)

12 program series Convince me, using Python, that series are real! (approximate f (x) = x 2 with series) f (x) = 1 a k=1 a k cos ( ) + bk sin ( ) Step 1 a 0 = 1 f (x)dx (6) Step 0. How can I compute the integral of a function in python. Objective: Compute the integral of x 2 in the range -2,2 import scipy.integrate as integrate # The return value is a tuple, with the first element # holding the estimated value of the integral # and the second element holding an upper bound error integrate.quad(function, start, end)

13 program series Convince me, using Python, that series are real! (approximate f (x) = x 2 with series) f (x) = 1 a k=1 a k cos ( ) + bk sin ( ) Step 1 a 0 = 1 f (x)dx (6) Step 0. How can I compute the integral of a function in python. Objective: Compute the integral of x 2 in the range -2,2 import scipy.integrate as integrate # The return value is a tuple, with the first element # holding the estimated value of the integral # and the second element holding an upper bound error integrate.quad(function, start, end) Step 0.5 Define your function f = x 2, your period = 2 and n the total number of terms that we want to use for the approximation.

14 program series Convince me, using Python, that series are real! (approximate f (x) = x 2 with series) f (x) = 1 a k=1 a k cos ( ) + bk sin ( ) Step 1 a 0 = 1 f (x)dx (6) Step 0. How can I compute the integral of a function in python. Objective: Compute the integral of x 2 in the range -2,2 import scipy.integrate as integrate # The return value is a tuple, with the first element # holding the estimated value of the integral # and the second element holding an upper bound error integrate.quad(function, start, end) Step 0.5 Define your function f = x 2, your period = 2 and n the total number of terms that we want to use for the approximation. Step 1 Compute the first coefficient a 0.

15 program series Approximate f (x) = x 2 with series Step 1.5 Make a function that receives x, k and computes f (x)cos(kx) import scipy.integrate as integrate # integrate.quad(f, start, end, args=(touple of parms)) integrate.quad(function, -1, 1, args=(k)) Step 2. Compute (using a loop or a comprehension): a k = 1 ( ) f (x)cos dx (7)

16 program series Approximate f (x) = x 2 with series Step 2.5 Make a function that receives x, k and computes f (x)sin(kx) Step 3. Compute (using a loop or a comprehension): b k = 1 f (x)sin Step 4. Compute the series: f (x) = 1 2 a 0 + a k cos k=1 ( ( kx ) dx (8) ) + b k sin ( ) (9)

17 program series Approximate f (x) = x 2 with series Step 2.5 Make a function that receives x, k and computes f (x)sin(kx) Step 3. Compute (using a loop or a comprehension): b k = 1 f (x)sin Step 4. Compute the series: f (x) = 1 2 a 0 + a k cos Step 5. Enjoy glory! k=1 ( ( kx ) dx (8) ) + b k sin ( ) (9)

18 Relations between cosine, sine and exponential functions and more i 2 = 1 x = a + bi defines a complex number e ±iθ = cos(θ) ± i sin(θ) cos(θ) = 1 (e +iθ + e iθ) 2 sin(θ) = 1 (e +iθ e iθ) 2i (10)