solving of the questions which are written in this file, are effective for your learning and reviewing. better files on the way. HG

1. In the name of God
Hossein Gholizadeh
Power engineering student
SBU

2. Find the Fourier series of below functions
𝑓 𝑥 = ቊ
𝑥 − 𝜋 < 𝑥 < 0
ℎ 0 < 𝑥 < 𝜋
𝑓 𝑥 = 𝑥 + sin 𝑥 − 𝜋 < 𝑥 < 𝜋
𝑓 𝑥 = 𝑒 𝑥 − 𝜋 < 𝑥 < 𝜋
𝑓 𝑥 = sin 𝑥 − 𝜋 < 𝑥 < 𝜋
𝑓 𝑥 = sin 𝑥 0 < 𝑥 < 𝜋

3. Find the sine Fourier series of below functions
𝑓 𝑥 = 𝑥2 0 < 𝑥 < 𝜋
𝑓 𝑥 = 𝑒 𝑥 0 < 𝑥 < 𝜋
𝑓 𝑥 = 𝑥 0 < 𝑥 < 𝑎

4. Find the cosine Fourier series of below functions
𝑓 𝑥 = cos ℎ𝑥 0 < 𝑥 < 𝜋
𝑓 𝑥 = 𝑥 0 < 𝑥 < 𝜋
𝑓 𝑥 = 𝑥 0 < 𝑥 < 𝑎

5. Find the Fourier series of below functions
𝑓 𝑥 = 𝑒−𝑥 0 < 𝑥 < 1
𝑓 𝑥 = sin ℎ𝑥 − 1 < 𝑥 < 1

6. Find the complex Fourier series of below functions
𝑓 𝑥 = 𝑒2𝑥 − 𝜋 < 𝑥 < 𝜋
𝑓 𝑥 = cos ℎ𝑥 − 𝜋 < 𝑥 < 𝜋

7. Find the Fourier series of below functions by derivation of other functions Fourier series
𝑓 𝑥 = 𝑠𝑖𝑛2 𝑥 0 < 𝑥 < 𝜋
𝑓 𝑥 = sin 𝑥 cos 𝑥 0 < 𝑥 < 𝜋

8. Find the Fourier series of functions by integration of below functions and series
෍
𝑘=1
∞
(−1) 𝑘+1
𝑘
sin 𝑘𝑥 =
𝑥
2
− 𝜋 < 𝑥 < 𝜋
4
𝜋
෍
𝑘=1
∞
sin 2𝑘 − 1 𝑥
2𝑘 − 1
= ቊ
−1 − 𝜋 < 𝑥 < 0
1 0 < 𝑥 < 𝜋
𝑥2 =
𝜋2
3
+ 4 ෍
𝑛=1
∞
(−1) 𝑛
𝑛2
𝑐𝑜𝑠𝑛𝑥 − 𝜋 < 𝑥 < 𝜋

9. According to written equation solve the written series.
𝜋
4
න
−𝜋
𝜋
(sin 𝑥)2 𝑑𝑥 = 2 + ෍
𝑘=1
∞
(1 + cos(2𝑘𝑥))2
( 2𝑘 2 − 1)2
1
12∗32 +
1
32∗52 +
1
52∗72+…

10. According to written equations, find answer of the question
𝑓 𝑡 = ൝
1 𝑡 < 1
0 𝑡 > 1
𝑓 𝑡 =
2
𝜋
න
0
∞
𝑠𝑖𝑛𝑤
𝑤
𝑐𝑜𝑠𝑤𝑡 𝑑𝑤
න
0
∞
𝑠𝑖𝑛𝑥2
𝑥
𝑑𝑥 =?
𝑓 𝑥 = න
0
∞
𝑎 𝑤 𝑐𝑜𝑠𝑤𝑥𝑑𝑤
𝑥𝑓 𝑥 = න
0
∞
𝐴 𝑤 𝑠𝑖𝑛𝑤𝑥𝑑𝑤 =?

11. • Is it possible to find Fourier integration for the below function?
𝑓 𝑥 = ቊ
0 𝑥 < 𝜋
−𝑠𝑖𝑛𝑥 𝑥 > 𝜋

12. Solve the below PDEs
𝑢 𝑥𝑥 − 𝑐2
𝑢 𝑡𝑡 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 𝑓 𝑥 0 < 𝑥 < 𝑙
𝑢 𝑡 𝑥, 0 = 𝑔 𝑥 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑙, 𝑡 = 0 𝑡 > 0

13. 𝑢 𝑥𝑥 − 𝑐2 𝑢 𝑡𝑡 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 𝑓 𝑥 0 < 𝑥 < 𝑙
𝑢 𝑡 𝑥, 0 = 0 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑙, 𝑡 = 0 𝑡 > 0

14. 𝑢 𝑥𝑥 − 𝑐2 𝑢 𝑡𝑡 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 0 0 < 𝑥 < 𝑙
𝑢 𝑡 𝑥, 0 = 𝑔 𝑥 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑙, 𝑡 = 0 𝑡 > 0

15. 𝑢 𝑥𝑥 − 𝑐2 𝑢 𝑡𝑡 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 𝑓 𝑥 0 < 𝑥 < 𝑙
𝑢 𝑡 𝑥, 0 = 0 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑥 𝑙, 𝑡 = 0 𝑡 > 0

16. 𝑢 𝑡𝑡 − 𝑐2 𝑢 𝑥𝑥 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 𝑓 𝑥 0 < 𝑥 < 𝑙
𝑢 𝑡 𝑥, 0 = 0 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 𝑥 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑥 𝑙, 𝑡 = 0 𝑡 > 0

17. 𝑢 𝑡𝑡 − 𝑐2 𝑢 𝑥𝑥 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 𝑓 𝑥 0 < 𝑥 < 𝑙
𝑢 𝑡 𝑥, 0 = 𝑔(𝑥) 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 𝑥 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑥 𝑙, 𝑡 = 0 𝑡 > 0

18. 𝑢 𝑥𝑥 − 𝑐2 𝑢 𝑡𝑡 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 0 0 < 𝑥 < 𝑙
𝑢 𝑡 𝑥, 0 = 𝑔(𝑥) 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 𝑥 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑥 𝑙, 𝑡 = 0 𝑡 > 0

19. 𝑢 𝑥𝑥 − 𝑐2 𝑢 𝑡𝑡 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 0 0 < 𝑥 < 𝑙
𝑢 𝑡 𝑥, 0 = 𝑔(𝑥) 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑥 𝑙, 𝑡 = 0 𝑡 > 0

20. 𝑢 𝑥𝑥 − 𝑐2 𝑢 𝑡𝑡 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 𝑓(𝑥) 0 < 𝑥 < 𝑙
𝑢 𝑡 𝑥, 0 = 𝑔(𝑥) 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑥 𝑙, 𝑡 = 0 𝑡 > 0

21. 𝑢 𝑡𝑡 + 𝑎𝑢 𝑡 + 𝑏𝑢 − 𝑐2 𝑢 𝑥𝑥 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 𝑓(𝑥) 0 < 𝑥 < 𝑙
𝑢 𝑡 𝑥, 0 = 0 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑥 𝑙, 𝑡 = 0 𝑡 > 0

22. 𝑢 𝑡𝑡 + 𝑎𝑢 𝑡 − 𝑐2 𝑢 𝑥𝑥 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 0 0 < 𝑥 < 𝑙
𝑢 𝑡 𝑥, 0 = 𝑔(𝑥) 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑥 𝑙, 𝑡 = 0 𝑡 > 0

23. 𝑢 𝑡 − 𝑐2 𝑢 𝑥𝑥 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠: 𝑢 𝑥, 0 = 𝑓(𝑥) 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑥 𝑙, 𝑡 = 0 𝑡 > 0

24. 𝑢 𝑡 − 𝑐2 𝑢 𝑥𝑥 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠: 𝑢 𝑥, 0 = 𝑓(𝑥) 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 𝑥 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑥 𝑙, 𝑡 = 0 𝑡 > 0

25. 𝑢 𝑡 − 𝑐2 𝑢 𝑥𝑥 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠: 𝑢 𝑥, 0 = 𝑓(𝑥) 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 𝑥 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑙, 𝑡 = 0 𝑡 > 0

26. 𝑢 𝑡 − 𝑐2 𝑢 𝑥𝑥 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠: 𝑢 𝑥, 0 = 𝑓(𝑥) 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 0 𝑡 > 0
𝑢 𝑙, 𝑡 = 0 𝑡 > 0

27. 𝑢 𝑡𝑡 − 𝑐2
𝑢 𝑥𝑥 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 𝑥 0 < 𝑥 < 1
𝑢 𝑡 𝑥, 0 = 0 0 < 𝑥 < 1
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 𝑡2
𝑡 > 0
𝑢 1, 𝑡 = 𝑐𝑜𝑠𝑡 𝑡 > 0
𝑢 𝑡𝑡 − 𝑐2
𝑢 𝑥𝑥 = ℎ(𝑥, 𝑡) 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 𝑓 𝑥 0 < 𝑥 < 𝑙
𝑢 𝑡 𝑥, 0 = 𝑔 𝑥 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 𝑥 0, 𝑡 = 𝑝(𝑡) 𝑡 > 0
𝑢 𝑥 𝑙, 𝑡 = 𝑞(𝑡) 𝑡 > 0

28. 𝑢 𝑡𝑡 − 𝑐2
𝑢 𝑥𝑥 = ℎ(𝑥) 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠 ቊ
𝑢 𝑥, 0 = 𝑓 𝑥 0 < 𝑥 < 𝑙
𝑢 𝑡 𝑥, 0 = 𝑔 𝑥 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 𝐴 𝑡 > 0
𝑢 𝑙, 𝑡 = 𝐵 𝑡 > 0
𝑢 𝑡 − 𝑐2 𝑢 𝑥𝑥 = 0 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠: 𝑢 𝑥, 0 = 𝑥(1 − 𝑥) 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 𝑡 𝑡 > 0
𝑢 𝑙, 𝑡 = 𝑠𝑖𝑛𝑡 𝑡 > 0

29. 𝑢 𝑡 − 𝑐2
𝑢 𝑥𝑥 = 𝑥𝑡 0 < 𝑥 < 1 𝑡 > 0
𝐼𝐶𝑠: 𝑢 𝑥, 0 = 𝑠𝑖𝑛𝜋𝑥 0 < 𝑥 < 1
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 𝑡 𝑡 > 0
𝑢 1, 𝑡 = 𝑡2 𝑡 > 0
𝑢 𝑡 − 𝑐2
𝑢 𝑥𝑥 = ℎ 0 < 𝑥 < 1 𝑡 > 0
𝐼𝐶𝑠: 𝑢 𝑥, 0 = 𝑢0 1 − 𝑐𝑜𝑠𝜋𝑥 0 < 𝑥 < 1
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 0 𝑡 > 0
𝑢 1, 𝑡 = 2 𝑢0 𝑡 > 0

30. 𝑢 𝑡 − 𝑐2
𝑢 𝑥𝑥 = 𝐴𝑒−𝑎𝑥
0 < 𝑥 < 𝜋 𝑡 > 0
𝐼𝐶𝑠: 𝑢 𝑥, 0 = 𝑠𝑖𝑛𝑥 0 < 𝑥 < 𝜋
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 0 𝑡 > 0
𝑢 𝜋, 𝑡 = 0 𝑡 > 0
𝑢 𝑡 − 𝑐2
𝑢 𝑥𝑥 = ℎ(𝑥, 𝑡) 0 < 𝑥 < 𝑙 𝑡 > 0
𝐼𝐶𝑠: 𝑢 𝑥, 0 = 𝑓(𝑥) 0 < 𝑥 < 𝑙
𝐵𝐶𝑠 ቊ
𝑢 0, 𝑡 = 𝑝(𝑡) 𝑡 > 0
𝑢 𝑙, 𝑡 = 𝑞(𝑡) 𝑡 > 0

31. 𝑢 𝑥𝑥 + 𝑢 𝑦𝑦 = 0 0 < 𝑥 < ∞ , 0 < 𝑦 < ∞
𝑢 𝑥, 0 = 𝑓 𝑥 0 < 𝑥 < ∞
𝑢 𝑥 0, 𝑦 = 𝑔 𝑦 0 < 𝑦 < ∞
lim
𝑥→∞
𝑢 𝑥, 𝑦 = 0
𝑢 𝑡 − 𝑢 𝑥𝑥 + ℎ 𝑡 𝑢 𝑥 = 𝛿 𝑥 𝛿 𝑡 0 < 𝑥 < ∞ 𝑡 > 0
𝑢 𝑥 0, 𝑡 = 𝑢 𝑥, 0 = 0
lim
𝑥→∞
𝑢 𝑥, 𝑡 = 0

32. 𝑢 𝑥𝑥 + 𝑢 𝑦𝑦 = 0 𝑥 > 0 0 < 𝑦 < 1
𝑢 𝑥, 0 = 𝑓 𝑥 𝑢 𝑥, 1 = 0 𝑥 > 0
𝑢 0, 𝑦 = 0,
lim
𝑦→∞
𝑢(𝑥, 𝑦) = 0
𝑢 𝑡𝑡 = 𝑐2 𝑢 𝑥𝑥 0 < 𝑥 < ∞ 𝑡 > 0
𝑢 𝑥, 0 = 𝑓 𝑥 𝑢 𝑡 𝑥, 0 = 0
𝑢 0, 𝑡 = 0 lim
𝑥→∞
𝑢(𝑥, 𝑡) = 0
𝑢 𝑡𝑡 = 𝑐2 𝑢 𝑥𝑥 0 < 𝑥 < 𝑙 𝑡 > 0
𝑢 𝑥, 0 = 0 𝑢 𝑡 𝑥, 0 = 0
𝑢 0, 𝑡 = 𝑓 𝑡 , 𝑢 𝑙, 𝑡 = 0 𝑡 > 0

33. 𝑢 𝑡 = 𝑘𝑢 𝑥𝑥 0 < 𝑥 < ∞ 𝑡 > 0
𝑢 𝑥, 0 = 𝑓0 0 < 𝑥 < ∞
𝑢 0, 𝑡 = 𝑓1 lim
𝑥→∞
𝑢(𝑥, 𝑡) = 𝑓0 𝑡 > 0
𝑢 𝑡 = 𝑘𝑢 𝑥𝑥 0 < 𝑥 < ∞ 𝑡 > 0
𝑢 𝑥, 0 = 𝑥 0 < 𝑥 < ∞
𝑢 0, 𝑡 = 0 lim
𝑥→∞
𝑢(𝑥, 𝑡) = 𝑥 𝑡 > 0
𝑢 𝑡 = 𝑘𝑢 𝑥𝑥 0 < 𝑥 < ∞ 𝑡 > 0
𝑢 𝑥, 0 = 𝑥 0 < 𝑥 < ∞
𝑢 0, 𝑡 = 𝑡2
lim
𝑥→∞
𝑢(𝑥, 𝑡) = 0 𝑡 > 0
𝑢 𝑡 = 𝑘𝑢 𝑥𝑥 − ℎ𝑢 0 < 𝑥 < ∞ 𝑡 > 0 ℎ 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑢 𝑥, 0 = 𝑓0 0 < 𝑥 < ∞
𝑢 0, 𝑡 = 0 lim
𝑥→∞
𝑢 𝑥(𝑥, 𝑡) = 0 𝑡 > 0

34. 𝑢 𝑡𝑡 = 𝑐2 𝑢 𝑥𝑥 0 < 𝑥 < ∞ 𝑡 > 0
𝑢 𝑥, 0 = 0 𝑢 𝑡 𝑥, 0 = 𝑓0
𝑢 0, 𝑡 = 0 lim
𝑥→∞
𝑢 𝑡(𝑥, 𝑡) = 0
𝑢 𝑡 = 𝑢 𝑥𝑥 + 𝑔 𝑥, 𝑡 0 < 𝑥 < 𝜋 𝑡 > 0
𝑢 𝑥, 0 = 𝑓 𝑥 0 < 𝑥 < 𝜋
𝑢 0, 𝑡 = 𝑢 𝜋, 𝑡 = 0
𝑢 𝑡 = 𝑢 𝑥𝑥 + 𝑔 𝑥, 𝑡 0 < 𝑥 < 𝜋 𝑡 > 0
𝑢 𝑥, 0 = 0 0 < 𝑥 < 𝜋
𝑢 0, 𝑡 = 0 𝑢 𝑥 𝜋, 𝑡 + ℎ𝑢 𝜋, 𝑡 = 0 𝑡 > 0
𝑢 𝑡 = 𝑢 𝑥𝑥 + 𝑔 𝑥, 𝑡 0 < 𝑥 < 𝜋 𝑡 > 0
𝑢 𝑥, 0 = 0 0 < 𝑥 < 𝜋
𝑢 0, 𝑡 = 𝑢 𝑥 𝜋, 𝑡 = 0

35. 𝑢 𝑡𝑡 = 𝑢 𝑥𝑥 + ℎ 0 < 𝑥 < 𝜋 𝑡 > 0
𝑢 𝑥, 0 = 0 𝑢 𝑡 𝑥, 0 = 0 0 < 𝑥 < 𝜋
𝑢 𝑥 0, 𝑡 = 𝑢 𝑥 𝜋, 𝑡 = 0
𝑢 𝑡𝑡 = 𝑢 𝑥𝑥 + 𝑔(𝑥) 0 < 𝑥 < 𝜋 𝑡 > 0
𝑢 𝑥, 0 = 0 𝑢 𝑡 𝑥, 0 = 0 0 < 𝑥 < 𝜋
𝑢 0, 𝑡 = 𝑢 𝜋, 𝑡 = 0

36. • Find the Fourier transform of the below functions
𝑓 𝑥 = 𝛿 𝑥
𝑓 𝑥 = 𝑢 𝑥
𝑓 𝑥 = 𝑋[−𝑎,𝑎] = ቊ
1 − 𝑎 < 𝑥 < 𝑎
0 𝑜𝑡ℎ𝑒𝑟𝑠
𝑓 𝑥 =
𝑠𝑖𝑛𝑥
𝑥
= 𝑠𝑖𝑛𝑐 𝑥

37. 𝑓 𝑥 = 𝑠𝑖𝑛𝑥
𝑓 𝑥 = 𝑐𝑜𝑠𝑥
𝑓 𝑥 = 𝑒−𝑎|𝑥|
𝑓 𝑥 = 𝑒−𝑎𝑥2
𝑓 𝑥 = 𝑒−𝑎𝑥 𝑢 𝑥
𝑓 𝑥 = 𝛿 𝑥 − 𝑎
𝑓 𝑥 = Λ(
𝑥
2𝑎
)

38. 𝑓 𝑥 = 𝑠𝑔𝑛 𝑥
𝑓 𝑥 = 𝛿′ 𝑥
𝑓 𝑥 = 𝛿(𝑥)(𝑛)
𝑓 𝑥 =
1
𝑥
𝑓 𝑥 = 𝑒 𝑗𝑎𝑥
𝑓 𝑥 = 𝑠𝑖𝑛𝑐2 𝑥
෍
−∞
∞
𝛿(𝑥 − 𝑛𝑇0)

39. • Prove the below equalities
න
−∞
∞
𝐹 𝑘 𝑔 𝑘 𝑒 𝑗𝑘𝑥
𝑑𝑘 = න
−∞
∞
𝑓 𝑦 𝐺 𝑦 − 𝑥 𝑑𝑦
𝑠𝑖𝑛𝑥 ∗ 𝑒−𝑎 𝑥
=
2
𝜋
𝑎𝑠𝑖𝑛𝑥
1 + 𝑎2
න
−∞
∞
𝐹 𝑘 𝑔 𝑘 𝑑𝑘 = න
−∞
∞
𝑓 𝑦 𝐺 𝑦 𝑑𝑦
න
0
∞
𝑒−𝑎2 𝑥2
𝑐𝑜𝑠𝑏𝑥𝑑𝑥 =
𝜋
2𝑎
𝑒
−𝑏2
4𝑎2
න
−∞
∞
(𝑓 ∗ 𝑔)(𝑥) = න
−∞
∞
𝑓 𝑢 𝑑𝑢 න
−∞
∞
𝑔 𝑣 𝑑𝑣

40. 𝑓 ∗ 𝑔 𝑚+𝑛 𝑥 = 𝑓 𝑚 𝑥 ∗ 𝑔 𝑛 (𝑥)
𝐹 𝑓 𝑎𝑥 − 𝑏 =
1
𝑎
𝑒
𝑗𝑘𝑏
𝑎 𝐹
𝑘
𝑎
Thanks for your attention