November 22, Center of Atmospheric Sciences, UNAM. Olmo S. Zavala Romero

Transcription

1 Center of Atmospheric Sciences, UNAM November 22, 2017

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 with respect to t 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 with respect to t 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) a 0 = 1 π a k = 1 π b k = 1 π k=1 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 with respect to x 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 ( kπx f (x)cos ( kx f (x)sin ( ) kπx + b k sin ) dx ) dx ( ) kπx (5)

7 Summary of numerical integration OPEN NEW SIDES!

8 program series Convince me, using Python, that series are real! Approximate f (x) = x 2 with series. f (x) = 1 2 a 0 + ( ) ( ) N k=1 a k cos kπx + b k sin kπx Step 1 Define your function f = x 2 (as a function), your period = 2 and N the total number of terms that we want to use for the approximation.

9 program series Convince me, using Python, that series are real! Approximate f (x) = x 2 with series. f (x) = 1 2 a 0 + ( ) ( ) N k=1 a k cos kπx + b k sin kπx Step 1 Define your function f = x 2 (as a function), your period = 2 and N the total number of terms that we want to use for the approximation. Step 2 How can we compute this first integral (do you remember numerical integration)? a 0 = 1 f (x)dx (6)

10 program series Convince me, using Python, that series are real! Approximate f (x) = x 2 with series. f (x) = 1 2 a 0 + ( ) ( ) N k=1 a k cos kπx + b k sin kπx Step 1 Define your function f = x 2 (as a function), your period = 2 and N the total number of terms that we want to use for the approximation. Step 2 How can we compute this first integral (do you remember numerical integration)? a 0 = 1 f (x)dx (6) Step 3 Compute the first coefficient a 0.

11 program series Convince me, using Python, that series are real! Approximate f (x) = x 2 with series. f (x) = 1 2 a 0 + ( ) ( ) N k=1 a k cos kπx + b k sin kπx Step 1 Define your function f = x 2 (as a function), your period = 2 and N the total number of terms that we want to use for the approximation. Step 2 How can we compute this first integral (do you remember numerical integration)? a 0 = 1 f (x)dx (6) Step 3 Compute the first coefficient a 0. Step 4. Compute simirarly (using a loop or a comprehension): a k = 1 ( ) kπx f (x)cos dx (7)

12 program series Convince me, using Python, that series are real! Approximate f (x) = x 2 with series. f (x) = 1 2 a 0 + ( ) ( ) N k=1 a k cos kπx + b k sin kπx Step 1 Define your function f = x 2 (as a function), your period = 2 and N the total number of terms that we want to use for the approximation. Step 2 How can we compute this first integral (do you remember numerical integration)? a 0 = 1 f (x)dx (6) Step 3 Compute the first coefficient a 0. Step 4. Compute simirarly (using a loop or a comprehension): a k = 1 ( ) kπx f (x)cos dx (7) Step 5. Compute the series: f (x) = 1 N ( ) ( ) kπx kπx 2 a 0 + a k cos + b k sin k=1 (8)

13 program series Show the example!

14 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 (9)