Page 245 - Handbook Of Integral Equations

3.5-3. Kernels Containing Tangent

b

17. tan(λx) – tan(λt) y(t) dt = f(x).

a
This is a special case of equation 3.8.3 with g(x) = tan(λx).
Solution:
1 d 2

y(x)= cos (λx)f (x) .
x
2λ dx
The right-hand side f(x) of the integral equation must satisfy certain relations (see item 2 of
◦
equation 3.8.3).
a

18. tan(βx) – tan(µt) y(t) dt = f(x), β >0, µ >0.

0
This is a special case of equation 3.8.4 with g(x) = tan(βx) and λ = µ/β.
b
k
19. tan x – tan t y(t) dt = f(x), 0 < k <1.
k
0
k
This is a special case of equation 3.8.3 with g(x) = tan x.
Solution:
1 d 2 k–1
y(x)= cos x cot xf (x) .

x
2k dx
◦
The right-hand side f(x) must satisfy certain conditions. As follows from item 3 of equation
3.8.3, the admissible general form of the right-hand side is given by

f(x)= F(x)+ Ax + B, A = –F (b), B = 1 2 bF (b) – F(0) – F(b) ,
x
x
where F(x) is an arbitrary bounded twice differentiable function (with bounded first deriva-
tive).
b
y(t)
20. dt = f(x), 0 < k <1.
a |tan(λx) – tan(λt)| k
This is a special case of equation 3.8.7 with g(x) = tan(λx)+β, where β is an arbitrary number.
a

21. k tan(λx) – t y(t) dt = f(x).

0
This is a special case of equation 3.8.5 with g(x)= k tan(λx).
a

22. x – k tan(λt) y(t) dt = f(x).

0
This is a special case of equation 3.8.6 with g(x)= k tan(λt).
3.5-4. Kernels Containing Cotangent
b

23. cot(λx) – cot(λt) y(t) dt = f(x).

a
This is a special case of equation 3.8.3 with g(x) = cot(λx).
b
k
24. cot x – cot t y(t) dt = f(x), 0 < k <1.
k
a
k
This is a special case of equation 3.8.3 with g(x) = cot x.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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