DET: Module 5&6

Question 1

Euler’s Formula

Answer 1

f(x) = a0/2 + ∑ancosnx + bnsinnx

a0 =1/π ∫f(x) dx
an =1/π ∫f(x) cosnx dx
bn =1/π ∫f(X) sinnx dx

limits: α to α+2π

Question 2

Half Range cosine series

Answer 2

f(x) = a0/2 + ∑ancos(nπx/l)

a0 = 2/l ∫f(x) dx
an = 2/l ∫f(x) cos(nπx/l) dx

limits: 0 to l

Question 3

Half Range sine series

Answer 3

f(x) = ∑bnsin(nπx/l)

bn = 2/l ∫f(x) sin(nπx/l) dx

limits: 0 to l

Question 4

Parseval’s Identity

Answer 4

1/L ∫[f(x)]^2 dx = a0^2/2 + ∑an^2 + bn^2

Question 5

∑1/n^2 =

Answer 5

π ^2 / 6

Question 6

Summation of 1(odd#)^2 : ∑1/(2n-1)^2 =

Answer 6

π ^2 / 8

Question 7

Fourier Transform

Answer 7

∫ f(x) e^isx dx

limits: -∞ to ∞

Question 8

Inverse Fourier Transform

Answer 8

1/2π ∫ f(x) e^-isx dx

limits: -∞ to ∞

Question 9

Fourier Cosine Transform

Answer 9

∫ f(x) cos sx dx

limits: 0 to ∞

Question 10

Fourier Sine Transform

Answer 10

∫ f(x) sin sx dx

limits: 0 to ∞

Question 11

Inverse Fourier Cosine Transform

Answer 11

2/π ∫ fc cos sx ds

limits: 0 to ∞

Question 12

Inverse Fourier Sine Transform

Answer 12

2/π ∫ fs sin sx ds

limits: 0 to ∞

Question 13

Finite Fourier Cosine Transform

Answer 13

∫ f(x) cos (nπx/c) dx

limits 0 to c

Question 14

Finite Fourier Sine Transform

Answer 14

∫ f(x) sin (nπx/c) dx

limits 0 to c

Question 15

e^i(ax) can be written as…

Answer 15

cos(ax) + isin(ax)

Question 16

Fc and Fs of e^-ax

Answer 16

Fc = sqrt(2/π) . a/a^2+s^2

Fs = sqrt(2/π) . s/a^2+s^2

Question 17

Fc and Fs of 1/x^2+a^2

Answer 17

Fc = sqrt(π/2) . e^-(as)/a

Fs = sqrt(π/2) . [1-e^-(as)]/a

Question 18

Fc and Fs of x/x^2+a^2

Answer 18

Fc = sqrt(π/2) . e^-(as)

Fs = sqrt(π/2) . e^-(as)

Question 19

∫ e^ax cosbx from 0 to ∞

Answer 19

a/a^2+b^2

Question 20

∫ e^ax sinbx from 0 to ∞

Answer 20

b/a^2+b^2