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smooth_function_of_a_linear_operator [2015/01/20 13:17] nikolaj |
smooth_function_of_a_linear_operator [2015/01/20 13:18] nikolaj |
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| $\dfrac{\partial}{\partial x}\,f(x)=\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\dfrac{\exp(h\frac{\partial}{\partial x})-1}{h}f(x)$ | $\dfrac{\partial}{\partial x}\,f(x)=\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\dfrac{\exp(h\frac{\partial}{\partial x})-1}{h}f(x)$ | ||
| - | $f(x)=\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\dfrac{\exp(h\frac{\partial}{\partial x})-1}{h}\int_0^x f(y)\,{\mathrm d}y$ | + | $f(x)=\lim_{h\to 0}\dfrac{\exp(h\frac{\partial}{\partial x})-1}{h}\int_0^x f(y)\,{\mathrm d}y$ |
| === References === | === References === | ||
