# Tagged Questions

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### solution of an integral equation in measurable functions

Let $\phi(t)$ be a positive continuous function on $[0,\infty)$ and $f(t,x)$ be a continuous function of two variables such that $$|f(t,x)|\leq \phi(t)|x|.$$ Suppose ...
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### Analyticity of Logarithmic Integrals

Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...
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### Prove existence and uniqueness of differential/integral equation

This is a homework question, so I'm essentially asking for hints and not answers due to academic honesty concerns. The course is in real analysis (baby rudin) and it is essentially chapter 9 which ...
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### How to prove $\int_a^b f(x)\varphi(x)dx=0\Rightarrow f(x)=0$

I am doing some reading on the calculus of variations and one of the first examples uses the following theorem: Let $f\in C[a,b]$. If $\int_a^b f(x)\varphi(x)dx=0$ for all $\varphi\in C[a,b]$, then ...
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### A convolution like integral equation

I would like to solve the following integral equation for $g(z)$. $$\int_z^\infty g(\zeta)(\zeta-z)^{\alpha-1} d\zeta = e^{-bz}, \tag{1}$$ where $\alpha$ and $b$ are constants. I would also like to ...
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### An integro differential equation involving $f$,$f_h$ and second derivative of $f$.

Let $f_h$ be the Hilbert transform of the real function $f$. I need some help solving this integro differential equation : $$\alpha f_h(x) + \beta f''(x) = f(x)$$ A simple sinusoid doesn't seem to ...
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### Integral equation corresponding to initial value problem

Question is to : Form an integral equation corresponding to the initial value problem $$\frac{d^2y}{dx^2}+y=0 ~; ~ x>0$$ with initial conditions $y(0)=1$ and $y'(0)=0$ What i have tried so far is ...
### Integral Equations $\phi(x), K(x,y)$
Suppose that $K(x,y)=g(x)h(y)$ and that $\int_a^b g(x)h(x)dx=0$. Let $\phi_0(x)=f(x))$. Show that all iterates equal the first iterate and find a simple formula for the solution. Basically, I ...