This document discusses Fourier integrals, which extend Fourier series to non-periodic functions. It defines the Fourier cosine integral and Fourier sine integral. As an example, it finds the Fourier cosine and sine integrals of the function f(x)=e-kx for x>0 and k>0. The Fourier cosine integral of this function is 2k/π∫0∞ cos(ωx)/(k2+ω2) dω and the Fourier sine integral is 2/π∫0∞ ωsin(ωx)/(k2+ω2) dω. In conclusion, Fourier integrals are used to find the Fourier representation of functions defined on the whole x-axis.
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A details explanation about Taylor's and Maclaurin's series with variety of examples are included in this slide. The aim is to give the viewer the basic knowledge about the topic.
Power Series,Taylor's and Maclaurin's SeriesShubham Sharma
A details explanation about Taylor's and Maclaurin's series with variety of examples are included in this slide. The aim is to give the viewer the basic knowledge about the topic.
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This document contains a couple of problems from the textbook for Calc 1, Boise State, Fall 2014. It also explains the table method for evaluating complicated derivatives
I am Grey N. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
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You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
I am Craig D. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from The University of Queensland. I have been helping students with their homework for the past 9 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com. You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
Lesson 12: Linear Approximations and Differentials (slides)Matthew Leingang
The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
This document contains a couple of problems from the textbook for Calc 1, Boise State, Fall 2014. It also explains the table method for evaluating complicated derivatives
I am Grey N. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
I am Craig D. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from The University of Queensland. I have been helping students with their homework for the past 9 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com. You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
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3. -: Fourier integrals :-
Fourier integral is an extension of Fourier series in non-periodic functions. Hear integration is used
instead of Summation in a Fourier series.
The Fourier integrals of a function f(x) is given by.
𝑓 𝑥 =
0
∞
𝐴 𝜔 cos 𝜔𝑥 + 𝐵 𝜔 𝑠𝑖𝑛𝜔𝑥 𝑑𝜔
Where,
𝐴 𝜔 =
1
𝜋 −∞
∞
𝑓 𝑢 cos 𝜔𝑢 𝑑𝑢
𝐵 𝜔 =
1
𝜋 −∞
∞
𝑓 𝑢 𝑠𝑖𝑛𝜔𝑢 𝑑𝑢
In this there are three type of integral.
I. Fourier integral
II. Fourier cosine integral
III. Fourier sine integral
4. -:Fourier cosine integral:-
Suppose f(x) is an even function .
As we know cos𝜔x is an even function and sin𝜔𝑥 is an odd function .There fore f(x) cos𝜔𝑥 is an
Even function & f(x) sin𝜔𝑥 is an odd function.
Now,
𝐴 𝜔 =
1
𝜋 −∞
∞
𝑓 𝑢 𝑐𝑜𝑠𝜔𝑢 𝑑𝑢 =
2
𝜋 0
∞
𝑓 𝑢 𝑐𝑜𝑠𝜔𝑢 𝑑𝑢
𝐵 𝜔 =
1
𝜋 −∞
∞
𝑓(𝑢) sin 𝜔𝑢 𝑑𝑢 = 0
Fourier cosine integral represented by
𝑓 𝑥 =
0
∞
𝐴(𝜔) cos 𝜔𝑥 𝑑𝜔
5. -:Fourier sine integral:-
Suppose f(x) is an even function. Now sin 𝜔𝑥 is an odd function then f(x) cos𝜔𝑥 is also odd function and
f(x) sin 𝜔𝑥 is an even function.
Now,
𝐴 𝜔 =
1
𝜋 −∞
∞
𝑓 𝑢 cos 𝜔𝑢 𝑑𝑢 = 0
𝐵 𝜔 =
1
𝜋 −∞
∞
𝑓 𝑢 sin 𝜔𝑢 𝑑𝑢 =
2
𝜋 0
∞
𝑓 𝑢 sin 𝜔𝑢 𝑑𝑢
Fourier sine integral represented as
𝑓 𝑥 =
0
∞
𝐵(𝜔) sin 𝜔𝑥 𝑑𝜔
6. Problem:-
Find the Fourier cosine and Fourier sine integral of 𝑓 𝑥 = 𝑒−𝑘𝑥
where x>0 and k>0.
Ans:-
Fourier cosine integral of f(x) is given by
𝑓 𝑥 =
0
∞
𝐴(𝜔) cos 𝜔𝑥 𝑑𝜔 … . . (1)
Where,
𝐴 𝜔 =
1
𝜋 −∞
∞
𝑓 𝑥 cos 𝜔𝑥 𝑑𝑥
=
1
𝜋 −∞
∞
𝑒−𝑘𝑥
cos 𝜔𝑥 𝑑𝑥
Since f(x) is even so the integration is even
=
2
𝜋 0
∞
𝑒−𝑘𝑥
cos 𝜔𝑥 𝑑𝑥
Now, by integration by parts
=
2
𝜋
−𝑘
𝑘2+𝜔2 𝑒−𝑘𝑥 −𝜔
𝑘
sin 𝜔𝑥 + cos 𝜔𝑥 ∞
0
=
2
𝜋
0 +
𝑘
𝑘2+𝜔2 =
2𝑘
𝜋 𝑘2+𝜔2
By substituting 𝐴 𝜔 into (1) we obtain the Fourier cosine integral
𝑓 𝑥 =
2𝑘
𝜋 0
∞
cos 𝜔𝑥
𝑘2 + 𝜔2
𝑑𝜔
7. Fourier sine integral of f(x) is given by
𝑓 𝑥 =
0
∞
𝐵(𝜔) sin 𝜔𝑥 𝑑𝜔 … … (2)
Where, 𝐵 𝜔 =
1
𝜋 −∞
∞
𝑓 𝑥 sin 𝜔𝑥 𝑑𝑥
Since f(x) is odd the integral is even
=
2
𝜋 0
∞
𝑒−𝑘𝑥
sin 𝜔𝑥 𝑑𝑥
Now, by integration by parts
=
2
𝜋
−𝜔
𝑘2 + 𝜔2
𝑒−𝑘𝑥
𝑘
𝜔
sin 𝜔𝑥 + cos 𝜔𝑥
∞
0
=
2
𝜋
0 +
𝜔
𝑘2 + 𝜔2
=
2
𝜋
𝜔
𝑘2 + 𝜔2
By substituting B 𝜔 into (2) we obtain the Fourier cosine integral
𝑓 𝑥 =
2
𝜋 0
∞
𝜔 sin 𝜔𝑥
𝑘2 + 𝜔2
𝑑𝜔
8. -:Conclusion:-
Many problems involve functions that are non –periodic and are of interest on the
whole x-axis to find Fourier series of such function we use Fourier integrals.