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3. Fredholm Integral Equations of the First Kind

3-1. Integral equations with kernels involving power-law functions

1. |x − t| y(t) dt = f(x).
2. |x − t|−1/2 y(t) dt = f(x).
3. |x − t|−k y(t) dt = f(x).
4. |_x_λ − t_λ|−_k y(t) dt = f(x).
5. |x − t|λ−1 y(t) dt = f(x).
6. sign(x − t) |x − t|λ−1 y(t) dt = f(x).
7. [a + b sign(x − t)] |x − t|λ−1 y(t) dt = f(x).
8. _t_−1[y(x + t) − y(x − t)] dt = f(x).
9. (t − x)−1 y(t) dt = f(x).
10. (t − x)−1 y(t) dt = f(x).

3-2. Integral equations with kernels involving exponential or logarithmic functions

1. _e_λ|x−t| y(t) dt = f(x).
2. ln|x − t| y(t) dt = f(x).Carleman's equation.
3. (ln|x − t| + β) y(t) dt = f(x).
4. ln (A/|x − t|) y(t) dt = f(x).
5. ln|(x + t)/(x − t)| y(t) dt = f(x).

3-3. Integral equations with kernels involving trigonometric functions

1. cos(xt) y(t) dt = f(x).
2. sin(xt) y(t) dt = f(x).
3. y(ξ) dt = f(x), ξ = x sin t.Schlomilch equation.
4. cot(t/2 − x/2) y(t) dt = f(x).

3-4. Other integral equations

1. _tJ_ν(xt) y(t) dt = f(x).
2. _K_0(|x − t|) y(t) dt = f(x).
3. K(x − t) y(t) dt = f(x).
4. K(x − t) y(t) dt = f(x).Wiener--Hopf equation of the first kind.

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