Paper 2, Section I, E

Further Complex Methods | Part II, 2010

Define

F±(x)=limϵ012πif(t)t(x±iϵ)dt,xRF^{\pm}(x)=\lim _{\epsilon \rightarrow 0} \frac{1}{2 \pi i} \int_{-\infty}^{\infty} \frac{f(t)}{t-(x \pm i \epsilon)} d t, \quad x \in \mathbb{R}

Using the fact that

F±(x)=±f(x)2+12πiPf(t)txdt,xRF^{\pm}(x)=\pm \frac{f(x)}{2}+\frac{1}{2 \pi i} P \int_{-\infty}^{\infty} \frac{f(t)}{t-x} d t, \quad x \in \mathbb{R}

where PP denotes the Cauchy principal value, find two complex-valued functions F+(z)F^{+}(z) and F(z)F^{-}(z) which satisfy the following conditions

  1. F+(z)F^{+}(z) and F(z)F^{-}(z) are analytic for Imz>0\operatorname{Im} z>0 and Imz<0\operatorname{Im} z<0 respectively, z=x+iyz=x+i y;

  2. F+(x)F(x)=sinxx,xRF^{+}(x)-F^{-}(x)=\frac{\sin x}{x}, \quad x \in \mathbb{R};

  3. F±(z)=O(1z),z,Imz0F^{\pm}(z)=\mathrm{O}\left(\frac{1}{z}\right), \quad z \rightarrow \infty, \quad \operatorname{Im} z \neq 0.

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