1
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0answers
80 views

Computing the logarithmic derivative of the numerator and denominator of a rational function.

Consider the rational function $R(z)=N(z)/D(z)$ where $N(z)$ and $D(z)$ are polynomials of $z$ with real coefficients. Furthermore, $N(0) \neq 0$, $D(0) \neq 0$, and $N(z)$ and $D(z)$ are relatively ...
1
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1answer
188 views

Using the estimation lemma

I have the question: Prove using the estimation lemma, for a function $f$ which is continuous in some region $D$ that: $\lim_{r \mapsto 0}\displaystyle\int_{\Sigma}\dfrac{f(z)}{(z-z_0)}\ dz = 2\pi ...
2
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1answer
252 views

How does this calculation of $\int_0^\infty(\sin^2x)/x^2\ dx$ work?

I'm having trouble with two steps in a calculation of $$\int_0^\infty\left(\frac{\sin x}{x}\right)^2\ dx$$ in a book. They take the contours $C_R$ composed of upper half-circles ...