Principal Value Integrals Principal Value Integrals Suppose f : [ a , c ) ∪ ( c , b ] → C f:[a, c)\cup (c,b]\rightarrow \mathbb C f : [ a , c ) ∪ ( c , b ] → C is integrable on [ a , c − ϵ ) ∪ ( c + ϵ , b ] [a, c-\epsilon)\cup (c+\epsilon, b] [ a , c − ϵ ) ∪ ( c + ϵ , b ] for all ϵ > 0 \epsilon > 0 ϵ > 0 . Define the principal value integral as
p . v . ∫ a b f ( x ) d x = lim ϵ → 0 ( ∫ a c − ϵ f ( x ) d x + ∫ c + ϵ b f ( x ) d x ) 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) p . v . ∫ a b f ( x ) d x = ϵ → 0 lim ( ∫ a c − ϵ f ( x ) d x + ∫ c + ϵ b f ( x ) d x ) Example Consider x − 1 x^{-1} x − 1 , which is not defined on x = 0 x=0 x = 0 , but
p . v . ∫ − 1 1 x − 1 d x = lim ϵ → 0 ( ∫ − 1 − ϵ x − 1 d x + ∫ ϵ 1 x − 1 d x ) = lim ϵ → 0 [ log ( x ) ] − 1 − ϵ + [ log ( x ) ] ϵ 1 = 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*} p . v . ∫ − 1 1 x − 1 d x = ϵ → 0 lim ( ∫ − 1 − ϵ x − 1 d x + ∫ ϵ 1 x − 1 d x ) = ϵ → 0 lim [ log ( x ) ] − 1 − ϵ + [ log ( x ) ] ϵ 1 = 0 Note that p . v . ∫ a b f ( x ) d x = ∫ a b f ( x ) d x p.v.\int_a^b f(x)dx = \int_a^b f(x)dx p . v . ∫ a b f ( x ) d x = ∫ a b f ( x ) d x if the integral exists. And the principal value integral at infinity is defined as
p . v . ∫ − ∞ ∞ f ( x ) d x = lim R → ∞ p . v . ∫ − R R f ( x ) d x = lim R → ∞ lim ϵ → 0 ( ∫ − R c − ϵ f ( x ) d x + ∫ c + R ∞ f ( x ) d x ) \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*} p . v . ∫ − ∞ ∞ f ( x ) d x = R → ∞ lim p . v . ∫ − R R f ( x ) d x = R → ∞ lim ϵ → 0 lim ( ∫ − R c − ϵ f ( x ) d x + ∫ c + R ∞ f ( x ) d x ) Solving Integrals with PVI Then, for some real functions, we can extend to complex, and take a way around its undefined point.
Lemma 1 lim z → z 0 ( z − z 0 ) f ( z ) = 0 ⟹ lim ϵ → 0 + ∫ C ( ϵ , θ 1 , θ 2 ) z 0 f ( z ) d z = 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 lim z → z 0 ( z − z 0 ) f ( z ) = 0 ⟹ lim ϵ → 0 + ∫ C ( ϵ , θ 1 , θ 2 ) z 0 f ( z ) d z = 0 .
proof . First, take ϵ \epsilon ϵ small enough s.t. z 0 z_0 z 0 is the only singularity in C ϵ C_\epsilon C ϵ . By residue theorem,
∫ C ϵ f ( z ) d z = 2 π i R e s ( f , z 0 ) = 2 π i lim z → z 0 ( z − z 0 ) 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 ∫ C ϵ f ( z ) d z = 2 πi R es ( f , z 0 ) = 2 πi z → z 0 lim ( z − z 0 ) f ( z ) = 0 Then, let a , b a, b a , b be the two endpoints of C ( ϵ , θ 1 , θ 2 ) z 0 C(\epsilon, \theta_1, \theta_2)_{z_0} C ( ϵ , θ 1 , θ 2 ) z 0 , so that C 1 = C ( ϵ , θ 1 , θ 2 ) z 0 C^{1} = C(\epsilon, \theta_1, \theta_2)_{z_0} C 1 = C ( ϵ , θ 1 , θ 2 ) z 0 is the upper arc of C ϵ C_{\epsilon} C ϵ , then, let C 2 C_{2} C 2 be the lower arc from a a a to b b b . Because ∫ C ϵ = 0 \int_{C_\epsilon} = 0 ∫ C ϵ = 0 , by deformation of curve, ∫ C 1 = ∫ C 2 \int_{C_{1} } = \int_{C_{2} } ∫ C 1 = ∫ C 2 . Also, note that ∫ C 2 + ( ∫ − C 1 ) = 0 \int_{C_2} + (\int_{-C_1}) = 0 ∫ C 2 + ( ∫ − C 1 ) = 0 so that ∫ C 1 = ∫ − C 1 = 0 \int_{C_1} = \int_{-C_1} = 0 ∫ C 1 = ∫ − C 1 = 0
Lemma 2 If f ( z ) f(z) f ( z ) has a pole of order 1 at z 0 z_0 z 0 then
lim ϵ → 0 + ∫ C ( ϵ , θ 1 , θ 2 ) z 0 f ( z ) d z = i ( θ 2 − θ 1 ) R e s ( f , z 0 ) \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) ϵ → 0 + lim ∫ C ( ϵ , θ 1 , θ 2 ) z 0 f ( z ) d z = i ( θ 2 − θ 1 ) R es ( f , z 0 ) proof . Consider the Laurent expansion of f f f ,
lim ϵ → 0 + ∫ C ( ϵ , θ 1 , θ 2 ) z 0 f ( z ) d z = lim ϵ → 0 + ∫ C ( ϵ , θ 1 , θ 2 ) z 0 ∑ n = − 1 ∞ c n ( z − z 0 ) n d z = lim ϵ → 0 + ∫ C ( ϵ , θ 1 , θ 2 ) z 0 c − 1 ( z − z 0 ) − 1 d z lemma 1 = lim ϵ → 0 + i ∫ θ 1 θ 2 c − 1 ( z 0 + ϵ e i t − z 0 ) − 1 ϵ e i t d t = i ( θ 2 − θ 1 ) c − 1 \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*} ϵ → 0 + lim ∫ C ( ϵ , θ 1 , θ 2 ) z 0 f ( z ) d z = ϵ → 0 + lim ∫ C ( ϵ , θ 1 , θ 2 ) z 0 n = − 1 ∑ ∞ c n ( z − z 0 ) n d z = ϵ → 0 + lim ∫ C ( ϵ , θ 1 , θ 2 ) z 0 c − 1 ( z − z 0 ) − 1 d z = ϵ → 0 + lim i ∫ θ 1 θ 2 c − 1 ( z 0 + ϵ e i t − z 0 ) − 1 ϵ e i t d t = i ( θ 2 − θ 1 ) c − 1 lemma 1 Example 1 Consider p . v . sin x x d x p.v. \frac{\sin^{x} }{x}dx p . v . x s i n x d x First note that
p . v . ∫ − ∞ ∞ e i x x d x = p . v . ∫ − ∞ ∞ cos x x d x + i p . v . ∫ − ∞ ∞ sin x x d x p.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 p . v . ∫ − ∞ ∞ x e i x d x = p . v . ∫ − ∞ ∞ x cos x d x + i p . v . ∫ − ∞ ∞ x sin x d x Because $\cos x / x $ is odd, $ p.v. \int_{-\infty}^\infty \frac{\cos x}{x}dx = 0$, we are left with
p . v . ∫ − ∞ ∞ sin x x d x = i − 1 p . v . ∫ − ∞ ∞ e i x x d x \:\:p.v.\int_{-\infty}^\infty \frac{\sin x}{x}dx = i^{-1}p.v. \int_{-\infty}^\infty \frac{e^{ix} }{x}dx p . v . ∫ − ∞ ∞ x sin x d x = i − 1 p . v . ∫ − ∞ ∞ x e i x d x Define f ( z ) = e i z / z f(z) = e^{iz}/z f ( z ) = e i z / z and consider the integrals
I C = lim R → ∞ lim ϵ → 0 ∮ C R , ϵ f ( z ) d z = lim R → ∞ lim ϵ → 0 ( ∫ C R , ϵ l i n e f ( z ) d z + ∫ C ( R , 0 , π ) f ( z ) d z − ∫ C ( ϵ , 0 , π ) f ( z ) d z ) = lim R → ∞ lim ϵ → 0 ∫ C R , ϵ l i n e f ( z ) d z + lim R → ∞ ∫ C ( R , 0 , π ) f ( z ) d z − lim ϵ → 0 ∫ C ( ϵ , 0 , π ) f ( z ) d z = I l i n e + I R − I ϵ \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*} I C = R → ∞ lim ϵ → 0 lim ∮ C R , ϵ f ( z ) d z = R → ∞ lim ϵ → 0 lim ( ∫ C R , ϵ l in e f ( z ) d z + ∫ C ( R , 0 , π ) f ( z ) d z − ∫ C ( ϵ , 0 , π ) f ( z ) d z ) = R → ∞ lim ϵ → 0 lim ∫ C R , ϵ l in e f ( z ) d z + R → ∞ lim ∫ C ( R , 0 , π ) f ( z ) d z − ϵ → 0 lim ∫ C ( ϵ , 0 , π ) f ( z ) d z = I l in e + I R − I ϵ Consider each integral, we have - By Residue Theorem, since I C I_C I C goes around the undefined point, I C = 0 I_C = 0 I C = 0 - By definition, I l i n e = lim R → ∞ lim ϵ → 0 p . v . ∫ − R R f ( z ) d z I_{line} = \lim_{R\rightarrow\infty}\lim_{\epsilon\rightarrow 0}p.v.\int_{-R}^R f(z)dz I l in e = lim R → ∞ lim ϵ → 0 p . v . ∫ − R R f ( z ) d z is exactly what we want. - By Jordon's Lemma, I R = 0 I_{R} = 0 I R = 0 . Thus, we are left with I l i n e = I ϵ = lim ϵ → 0 ∫ C ( ϵ , 0 , π ) f ( z ) d z I_{line} = I_\epsilon = \lim_{\epsilon\rightarrow 0}\int_{C(\epsilon,0,\pi)} f(z)dz I l in e = I ϵ = lim ϵ → 0 ∫ C ( ϵ , 0 , π ) f ( z ) d z is what we want. By lemma 2,
lim ϵ → 0 ∫ C ( ϵ , 0 , π ) f ( z ) d z = i ( π − 0 ) R e s ( f , 0 ) = i π ( lim z → 0 z e i z z ) = i π \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 ϵ → 0 lim ∫ C ( ϵ , 0 , π ) f ( z ) d z = i ( π − 0 ) R es ( f , 0 ) = iπ ( z → 0 lim z z e i z ) = iπ Therefore,
p . v . sin x x d x = i − 1 I l i n e = π p.v. \frac{\sin x }{x}dx = i^{-1}I_{line} = \pi p . v . x sin x d x = i − 1 I l in e = π Example 2 Similarly, if the principal value integral has multiple undefined points, we can go around each of them. Consider p . v . ∫ − ∞ ∞ ( x 2 − 1 ) d x p.v. \int_{-\infty}^\infty (x^2 - 1) dx p . v . ∫ − ∞ ∞ ( x 2 − 1 ) d x
Similarly, I C = I R + I l i n e − I ϵ , − 1 − I ϵ , 1 I_C = I_{R} + I_{line} - I_{\epsilon, -1} - I_{\epsilon, 1} I C = I R + I l in e − I ϵ , − 1 − I ϵ , 1 - I C = 0 I_C = 0 I C = 0 by residue theorem - I l i n e = lim R → ∞ lim ϵ → 0 ∫ C R , ϵ l i n e ( z 2 − 1 ) − 1 d z = p . v . ∫ − ∞ ∞ ( x 2 − 1 ) − 1 d x I_{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 I l in e = lim R → ∞ lim ϵ → 0 ∫ C R , ϵ l in e ( z 2 − 1 ) − 1 d z = p . v . ∫ − ∞ ∞ ( x 2 − 1 ) − 1 d x is what we want - $ I_{\epsilon, -1} + I_{\epsilon, 1} = I_{line}$ is what we calculate. By lemma 2, since z = − 1 z=-1 z = − 1 and z = 1 z=1 z = 1 are both poles of order 1,
I ϵ , 1 = π i R e s ( f , 1 ) = lim z → 1 ( z + 1 ) − 1 = π i / 2 I_{\epsilon, 1} = \pi i Res(f, 1) = \lim_{z\rightarrow 1}(z+1)^{-1} = \pi i /2 I ϵ , 1 = πi R es ( f , 1 ) = z → 1 lim ( z + 1 ) − 1 = πi /2 I ϵ , − 1 = π i R e s ( f , − 1 ) = lim z → − 1 ( z − 1 ) − 1 = − π i / 2 I_{\epsilon, -1} = \pi i Res(f, -1) = \lim_{z\rightarrow -1}(z-1)^{-1} = -\pi i/2 I ϵ , − 1 = πi R es ( f , − 1 ) = z → − 1 lim ( z − 1 ) − 1 = − πi /2 ∫ − ∞ ∞ ( x 2 − 1 ) − 1 d x = π i / 2 − π i / 2 = 0 \int_{-\infty}^\infty (x^2-1)^{-1}dx = \pi i /2 - \pi i /2 = 0 ∫ − ∞ ∞ ( x 2 − 1 ) − 1 d x = πi /2 − πi /2 = 0 Integrals with Branch Cuts Consider functions on C \mathbb C C with branch points at 0 0 0 , and we only defined the funciton on some domain D = { r e i θ : r > 0 , θ ∈ ( 0 , 2 π ) } D = \{re^i\theta: r> 0, \theta\in (0, 2\pi)\} D = { r e i θ : r > 0 , θ ∈ ( 0 , 2 π )} with a branch cut.
For function f ( z ) f(z) f ( z ) , define f + , f − : ( 0 , ∞ ) → C f_+, f_-: (0,\infty)\rightarrow\mathbb C f + , f − : ( 0 , ∞ ) → C as
f + ( r ) = lim θ → 0 + f ( r e i θ ) , f − ( r ) = lim θ → 2 π − f ( r e i θ ) f_+(r) = \lim_{\theta\rightarrow 0^+} f(re^{i\theta}), f_-(r) = \lim_{\theta\rightarrow 2\pi^-} f(re^{i\theta}) f + ( r ) = θ → 0 + lim f ( r e i θ ) , f − ( r ) = θ → 2 π − lim f ( r e i θ ) Example 1 Consider the sqrt function f ( r e i θ ) = r 1 / 2 e i θ / 2 f(re^{i\theta}) = r^{1/2} e^{i\theta/2} f ( r e i θ ) = r 1/2 e i θ /2 , then
f + ( r ) = r 1 / 2 , f − ( r ) = − r 1 / 2 f_+(r) = r^{1/2}, f_-(r) = -r^{1/2} f + ( r ) = r 1/2 , f − ( r ) = − r 1/2 Example 2 evaluate ∫ 0 ∞ x x 3 + 1 d x \int_0^\infty \frac{\sqrt x}{x^3+1}dx ∫ 0 ∞ x 3 + 1 x d x
First, note that
lim R → ∞ lim ϵ → 0 lim θ → 0 + I C = lim R → ∞ I C R − lim ϵ → 0 I C ϵ + lim R → ∞ lim ϵ → 0 lim θ → 0 + I C R , ϵ + + lim R → ∞ lim ϵ → 0 lim θ → 0 + I C R , 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}} R → ∞ lim ϵ → 0 lim θ → 0 + lim I C = R → ∞ lim I C R − ϵ → 0 lim I C ϵ + R → ∞ lim ϵ → 0 lim θ → 0 + lim I C R , ϵ + + R → ∞ lim ϵ → 0 lim θ → 0 + lim I C R , ϵ − Then, note that lim z → ∞ z f ( z ) = lim z → ∞ z 3 / 2 z 3 + 1 = 0 , lim z → 0 z f ( z ) = 0 3 / 2 0 3 + 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 lim z → ∞ z f ( z ) = lim z → ∞ z 3 + 1 z 3/2 = 0 , lim z → 0 z f ( z ) = 0 3 + 1 0 3/2 = 0 , so that
lim R → ∞ I C R = 0 , lim ϵ → 0 I C ϵ = 0 \lim_{R\rightarrow\infty}I_{C_R} = 0, \lim_{\epsilon\rightarrow 0}I_{C_\epsilon} = 0 R → ∞ lim I C R = 0 , ϵ → 0 lim I C ϵ = 0 Then, consider the left two terms
lim R → ∞ lim ϵ → 0 lim θ → 0 + I C R , ϵ + = lim R → ∞ lim ϵ → 0 ∫ ϵ R f + ( x ) d x = ∫ 0 ∞ f + ( x ) d x \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 R → ∞ lim ϵ → 0 lim θ → 0 + lim I C R , ϵ + = R → ∞ lim ϵ → 0 lim ∫ ϵ R f + ( x ) d x = ∫ 0 ∞ f + ( x ) d x lim R → ∞ lim ϵ → 0 lim θ → 0 + I C R , ϵ − = lim R → ∞ lim ϵ → 0 ∫ R ϵ f − ( z ) d z = − ∫ 0 ∞ f − ( z ) d z \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 R → ∞ lim ϵ → 0 lim θ → 0 + lim I C R , ϵ − = R → ∞ lim ϵ → 0 lim ∫ R ϵ f − ( z ) d z = − ∫ 0 ∞ f − ( z ) d z via branch cut of square root functions, we have − f − ( z ) = f + ( z ) = f ( x ) -f_-(z) = f_+(z) = f(x) − f − ( z ) = f + ( z ) = f ( x ) so that the whole integral
lim R → ∞ lim ϵ → 0 lim θ → 0 + I C = ∫ 0 ∞ f + ( x ) − ∫ 0 ∞ f − ( x ) = 2 ∫ 0 ∞ x x 3 + 1 d x \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 R → ∞ lim ϵ → 0 lim θ → 0 + lim I C = ∫ 0 ∞ f + ( x ) − ∫ 0 ∞ f − ( x ) = 2 ∫ 0 ∞ x 3 + 1 x d x Now, using residue theorem,
I C = ∫ C f ( z ) d z = 2 π i ( R e s ( f , − 1 ) + R e s ( f , e i π / 3 ) + R e s ( f , e − i π / 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})) I C = ∫ C f ( z ) d z = 2 πi ( R es ( f , − 1 ) + R es ( f , e iπ /3 ) + R es ( f , e − iπ /3 )) so that the original integral
∫ 0 ∞ x x 3 + 1 d x = π i ( R e s ( f , − 1 ) + R e s ( f , e i π / 3 ) + R e s ( f , e − i π / 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})) ∫ 0 ∞ x 3 + 1 x d x = πi ( R es ( f , − 1 ) + R es ( f , e iπ /3 ) + R es ( f , e − iπ /3 )) January 11, 2025 January 9, 2023