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Infinite Products and Weierstrass Factorization
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Tagged analysis, complex analysis, series
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Complex analysis notes
Cauchy-Riemann Let $f(z) = f(x + iy) = g(x, y) + ih(x, y)$. If it’s holomorphic, we can differentiate along the real line and the imaginary line: $f’ = \lim\frac{f(x + a, y) – f(x, y)}{a} = \frac{∂g}{∂x} + i\frac{∂h}{∂x}$ $f’ = … Continue reading
Uniqueness of Laplace transform
Statement: Let $L_f(t) = \int_0^\infty f(x)e^{-tx} dx$. Then if $f(x) = o(e^{ax})$ for some $a > 0$, $L_f = L_g$ implies $f = g$. Proof: By linearity of the integral, this is equivalent to showing $L_f = 0$ implies $f = … Continue reading
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Tagged analysis, laplace transform
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Convergence theorems
Monotone Convergence Theorem Statement: Suppose we have a sequence of non-negative functions $f_n$ that converges almost everywhere to $f$ on some set $D$ on a measure space. Then $\lim \int_D f_n d\mu = \int_D f d\mu$. Proof: Since the sequence of … Continue reading