mathjacks
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1,201
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### Questions (139)

 7 Describe the Riemann surface for $w=z^2-1$. 5 Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$. 5 Show the level curves of $\log|f(z)|$ are orthogonal to those of $\operatorname{arg}(f(z))$. 5 Prove $(\overline{A \cap B}) \subseteq \overline{A} \cap \overline{B}$. 5 Compute $\lim_{x \to 0} \frac{\sin(x)+\cos(x)-e^x}{\log(1+x^2)}$.

### Reputation (1,201)

 +15 Why does Fubini's theorem not hold/apply to this function? +25 Let $1 \leq p <\infty$ and $f \in L^p(\mathbb{R})$. Prove $\lim_{x \to \infty} \int_x^{x+1} f(t) dt = 0$. +5 Show $(\partial^2 z / \partial x \partial y)^2 = \frac{\partial^2z}{\partial x^2} \cdot \frac{\partial^2z}{\partial y^2}$ for $z=ax + yf(a)+\phi(a)$. +5 Demand $z=x+y$ and $x^2/4 + y^2/5 + z^2/25 = 1$. What is the maximum value of $f(x,y,z) = x^2+y^2+z^2$?

### Tags (58)

 0 real-analysis × 48 0 calculus × 11 0 complex-analysis × 38 0 lebesgue-integral × 10 0 proof-verification × 18 0 sequences-and-series × 10 0 multivariable-calculus × 16 0 proof-strategy × 10 0 lebesgue-measure × 14 0 elementary-set-theory × 7

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