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Principal Value Integrals

Principal Value Integrals

Suppose f:[a,c)(c,b]Cf:[a, c)\cup (c,b]\rightarrow \mathbb C is integrable on [a,cϵ)(c+ϵ,b][a, c-\epsilon)\cup (c+\epsilon, b] for all ϵ>0\epsilon > 0. Define the principal value integral as

p.v.abf(x)dx=limϵ0(acϵf(x)dx+c+ϵbf(x)dx)p.v.\int_a^b f(x)dx = \lim_{\epsilon\rightarrow 0} \big(\int_a^{c-\epsilon} f(x)dx + \int_{c+\epsilon}^b f(x)dx\big)

Example Consider x1x^{-1}, which is not defined on x=0x=0, but

p.v.11x1dx=limϵ0(1ϵx1dx+ϵ1x1dx)=limϵ0[log(x)]1ϵ+[log(x)]ϵ1=0\begin{align*} p.v. \int_{-1}^1 x^{-1}dx &= \lim_{\epsilon\rightarrow 0} \big(\int_{-1}^{-\epsilon} x^{-1}dx + \int_{\epsilon}^1 x^{-1}dx\big) \\ &= \lim_{\epsilon\rightarrow 0} [\log(x)]^{-\epsilon}_{-1} + [\log(x)]^{1}_{\epsilon}\\ &= 0 \end{align*}

Note that p.v.abf(x)dx=abf(x)dxp.v.\int_a^b f(x)dx = \int_a^b f(x)dx if the integral exists.
And the principal value integral at infinity is defined as

p.v.f(x)dx=limRp.v.RRf(x)dx=limRlimϵ0(Rcϵf(x)dx+c+Rf(x)dx)\begin{align*} p.v.\int_{-\infty}^\infty f(x)dx &= \lim_{R\rightarrow\infty}p.v.\int_{-R}^R f(x)dx\\ &=\lim_{R\rightarrow\infty} \lim_{\epsilon\rightarrow 0} \big(\int_{-R}^{c-\epsilon} f(x)dx + \int_{c+R}^\infty f(x)dx\big) \end{align*}

Solving Integrals with PVI

Then, for some real functions, we can extend to complex, and take a way around its undefined point.

Lemma 1

limzz0(zz0)f(z)=0    limϵ0+C(ϵ,θ1,θ2)z0f(z)dz=0\lim_{z\rightarrow z_0}(z-z_0)f(z) = 0\implies \lim_{\epsilon\rightarrow 0^+} \int_{C(\epsilon, \theta_1, \theta_2)_{z_0} }f(z)dz = 0.

proof. First, take ϵ\epsilon small enough s.t. z0z_0 is the only singularity in CϵC_\epsilon. By residue theorem,

Cϵf(z)dz=2πiRes(f,z0)=2πilimzz0(zz0)f(z)=0\int_{C_\epsilon} f(z)dz = 2\pi i Res(f, z_0) = 2\pi i \lim_{z\rightarrow z_0}(z-z_0)f(z) = 0

Then, let a,ba, b be the two endpoints of C(ϵ,θ1,θ2)z0C(\epsilon, \theta_1, \theta_2)_{z_0}, so that C1=C(ϵ,θ1,θ2)z0C^{1} = C(\epsilon, \theta_1, \theta_2)_{z_0} is the upper arc of CϵC_{\epsilon}, then, let C2C_{2} be the lower arc from aa to bb. Because Cϵ=0\int_{C_\epsilon} = 0, by deformation of curve, C1=C2\int_{C_{1} } = \int_{C_{2} }. Also, note that C2+(C1)=0\int_{C_2} + (\int_{-C_1}) = 0 so that C1=C1=0\int_{C_1} = \int_{-C_1} = 0

Lemma 2

If f(z)f(z) has a pole of order 1 at z0z_0 then

limϵ0+C(ϵ,θ1,θ2)z0f(z)dz=i(θ2θ1)Res(f,z0)\lim_{\epsilon\rightarrow 0^+} \int_{C(\epsilon, \theta_1, \theta_2)_{z_0} }f(z)dz = i(\theta_2 - \theta_1)Res(f, z_0)

proof. Consider the Laurent expansion of ff,

limϵ0+C(ϵ,θ1,θ2)z0f(z)dz=limϵ0+C(ϵ,θ1,θ2)z0n=1cn(zz0)ndz=limϵ0+C(ϵ,θ1,θ2)z0c1(zz0)1dzlemma 1=limϵ0+iθ1θ2c1(z0+ϵeitz0)1ϵeitdt=i(θ2θ1)c1\begin{align*} \lim_{\epsilon\rightarrow 0^+} \int_{C(\epsilon, \theta_1, \theta_2)_{z_0} }f(z)dz &= \lim_{\epsilon\rightarrow 0^+} \int_{C(\epsilon, \theta_1, \theta_2)_{z_0} }\sum_{n=-1}^\infty c_n (z-z_0)^ndz\\ &= \lim_{\epsilon\rightarrow 0^+} \int_{C(\epsilon, \theta_1, \theta_2)_{z_0} } c_{-1} (z-z_0)^{-1}dz&\text{lemma 1}\\ &= \lim_{\epsilon\rightarrow 0^+}i\int_{\theta_1}^{\theta_2} c_{-1} (z_0 + \epsilon e^{it} - z_0)^{-1} \epsilon e^{it}dt\\ &= i(\theta_2 - \theta_1) c_{-1} \end{align*}

Example 1

Consider p.v.sinxxdxp.v. \frac{\sin^{x} }{x}dx First note that

p.v.eixxdx=p.v.cosxxdx+i  p.v.sinxxdxp.v. \int_{-\infty}^\infty \frac{e^{ix} }{x}dx = p.v. \int_{-\infty}^\infty \frac{\cos x}{x}dx + i \:\:p.v.\int_{-\infty}^\infty \frac{\sin x}{x}dx

Because $\cos x / x $ is odd, $ p.v. \int_{-\infty}^\infty \frac{\cos x}{x}dx = 0$, we are left with

  p.v.sinxxdx=i1p.v.eixxdx \:\:p.v.\int_{-\infty}^\infty \frac{\sin x}{x}dx = i^{-1}p.v. \int_{-\infty}^\infty \frac{e^{ix} }{x}dx

Define f(z)=eiz/zf(z) = e^{iz}/z and consider the integrals

IC=limRlimϵ0CR,ϵf(z)dz=limRlimϵ0(CR,ϵlinef(z)dz+C(R,0,π)f(z)dzC(ϵ,0,π)f(z)dz)=limRlimϵ0CR,ϵlinef(z)dz+limRC(R,0,π)f(z)dzlimϵ0C(ϵ,0,π)f(z)dz=Iline+IRIϵ\begin{align*} I_C &= \lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0} \oint_{C_{R,\epsilon} } f(z)dz\\ &= \lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0} (\int_{C_{R, \epsilon}^{line} } f(z)dz + \int_{C(R,0,\pi)} f(z)dz - \int_{C(\epsilon,0,\pi)} f(z)dz )\\ &= \lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0} \int_{C_{R, \epsilon}^{line} } f(z)dz + \lim_{R\rightarrow\infty}\int_{C(R,0,\pi)} f(z)dz - \lim_{\epsilon\rightarrow 0}\int_{C(\epsilon,0,\pi)} f(z)dz \\ &= I_{line} + I_R - I_\epsilon \end{align*}

Consider each integral, we have - By Residue Theorem, since ICI_C goes around the undefined point, IC=0I_C = 0
- By definition, Iline=limRlimϵ0p.v.RRf(z)dzI_{line} = \lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0}p.v.\int_{-R}^R f(z)dz is exactly what we want.
- By Jordon's Lemma, IR=0I_{R} = 0.
Thus, we are left with Iline=Iϵ=limϵ0C(ϵ,0,π)f(z)dzI_{line} = I_\epsilon = \lim_{\epsilon\rightarrow 0}\int_{C(\epsilon,0,\pi)} f(z)dz is what we want.
By lemma 2,

limϵ0C(ϵ,0,π)f(z)dz=i(π0)Res(f,0)=iπ(limz0zeizz)=iπ\lim_{\epsilon\rightarrow 0}\int_{C(\epsilon,0,\pi)} f(z)dz = i(\pi - 0) Res(f, 0) = i\pi (\lim_{z\rightarrow 0} z \frac{e^{iz} }{z}) = i\pi

Therefore,

p.v.sinxxdx=i1Iline=πp.v. \frac{\sin x }{x}dx = i^{-1}I_{line} = \pi

Example 2

Similarly, if the principal value integral has multiple undefined points, we can go around each of them. Consider p.v.(x21)dxp.v. \int_{-\infty}^\infty (x^2 - 1) dx

Similarly, IC=IR+IlineIϵ,1Iϵ,1I_C = I_{R} + I_{line} - I_{\epsilon, -1} - I_{\epsilon, 1} - IC=0I_C = 0 by residue theorem - Iline=limRlimϵ0CR,ϵline(z21)1dz=p.v.(x21)1dxI_{line} = \lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0} \int_{C_{R, \epsilon}^{line} } (z^2 - 1)^{-1}dz = p.v. \int_{-\infty}^\infty (x^2-1)^{-1}dx is what we want - $ I_{\epsilon, -1} + I_{\epsilon, 1} = I_{line}$ is what we calculate. By lemma 2, since z=1z=-1 and z=1z=1 are both poles of order 1,

Iϵ,1=πiRes(f,1)=limz1(z+1)1=πi/2I_{\epsilon, 1} = \pi i Res(f, 1) = \lim_{z\rightarrow 1}(z+1)^{-1} = \pi i /2
Iϵ,1=πiRes(f,1)=limz1(z1)1=πi/2I_{\epsilon, -1} = \pi i Res(f, -1) = \lim_{z\rightarrow -1}(z-1)^{-1} = -\pi i/2
(x21)1dx=πi/2πi/2=0 \int_{-\infty}^\infty (x^2-1)^{-1}dx = \pi i /2 - \pi i /2 = 0

Integrals with Branch Cuts

Consider functions on C\mathbb C with branch points at 00, and we only defined the funciton on some domain D={reiθ:r>0,θ(0,2π)}D = \{re^i\theta: r> 0, \theta\in (0, 2\pi)\} with a branch cut.

For function f(z)f(z), define f+,f:(0,)Cf_+, f_-: (0,\infty)\rightarrow\mathbb C as

f+(r)=limθ0+f(reiθ),f(r)=limθ2πf(reiθ)f_+(r) = \lim_{\theta\rightarrow 0^+} f(re^{i\theta}), f_-(r) = \lim_{\theta\rightarrow 2\pi^-} f(re^{i\theta})

Example 1

Consider the sqrt function f(reiθ)=r1/2eiθ/2f(re^{i\theta}) = r^{1/2} e^{i\theta/2}, then

f+(r)=r1/2,f(r)=r1/2f_+(r) = r^{1/2}, f_-(r) = -r^{1/2}

Example 2

evaluate 0xx3+1dx\int_0^\infty \frac{\sqrt x}{x^3+1}dx

First, note that

limRlimϵ0limθ0+IC=limRICRlimϵ0ICϵ+limRlimϵ0limθ0+ICR,ϵ++limRlimϵ0limθ0+ICR,ϵ\lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0}\lim_{\theta\rightarrow0^+}I_C = \lim_{R\rightarrow\infty}I_{C_R} - \lim_{\epsilon\rightarrow 0}I_{C_\epsilon} + \lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0}\lim_{\theta\rightarrow0^+} I_{C^{+}_{R,\epsilon}} + \lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0}\lim_{\theta\rightarrow0^+} I_{C^{-}_{R,\epsilon}}

Then, note that limzzf(z)=limzz3/2z3+1=0,limz0zf(z)=03/203+1=0\lim_{z\rightarrow\infty} z f(z) = \lim_{z\rightarrow\infty}\frac{z^{3/2}}{z^3+1} = 0, \lim_{z\rightarrow 0} z f(z) = \frac{0^{3/2}}{0^3+1} = 0, so that

limRICR=0,limϵ0ICϵ=0\lim_{R\rightarrow\infty}I_{C_R} = 0, \lim_{\epsilon\rightarrow 0}I_{C_\epsilon} = 0

Then, consider the left two terms

limRlimϵ0limθ0+ICR,ϵ+=limRlimϵ0ϵRf+(x)dx=0f+(x)dx\lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0}\lim_{\theta\rightarrow0^+} I_{C^{+}_{R,\epsilon}} = \lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0}\int_{\epsilon}^R f_+(x)dx = \int_{0}^\infty f_+(x)dx
limRlimϵ0limθ0+ICR,ϵ=limRlimϵ0Rϵf(z)dz=0f(z)dz\lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0}\lim_{\theta\rightarrow0^+} I_{C^{-}_{R,\epsilon}} = \lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0}\int_{R}^{\epsilon} f_-(z)dz = -\int_{0}^\infty f_-(z)dz

via branch cut of square root functions, we have f(z)=f+(z)=f(x)-f_-(z) = f_+(z) = f(x) so that the whole integral

limRlimϵ0limθ0+IC=0f+(x)0f(x)=20xx3+1dx\lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0}\lim_{\theta\rightarrow0^+}I_C = \int_{0}^\infty f_+(x) -\int_{0}^\infty f_-(x) = 2 \int_0^\infty \frac{\sqrt x }{x^3+1} dx

Now, using residue theorem,

IC=Cf(z)dz=2πi(Res(f,1)+Res(f,eiπ/3)+Res(f,eiπ/3))I_C = \int_{C} f(z)dz = 2\pi i (Res(f, -1) + Res(f, e^{i\pi/3}) + Res(f, e^{-i\pi/3}))

so that the original integral

0xx3+1dx=πi(Res(f,1)+Res(f,eiπ/3)+Res(f,eiπ/3))\int_0^\infty \frac{\sqrt x }{x^3+1} dx = \pi i (Res(f, -1) + Res(f, e^{i\pi/3}) + Res(f, e^{-i\pi/3}))