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Does anyone know precisely what causes phantom scrollbars in <math display=block>? [ edit ]
1.
J
−
1
=
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x
r
y
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z
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r
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)
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{\displaystyle 1.\qquad J^{-1}={\begin{pmatrix}{\dfrac {x}{r}}&{\dfrac {y}{r}}&{\dfrac {z}{r}}\\\\{\dfrac {xz}{r^{2}{\sqrt {x^{2}+y^{2}}}}}&{\dfrac {yz}{r^{2}{\sqrt {x^{2}+y^{2}}}}}&{\dfrac {-(x^{2}+y^{2})}{r^{2}{\sqrt {x^{2}+y^{2}}}}}\\\\{\dfrac {-y}{x^{2}+y^{2}}}&{\dfrac {x}{x^{2}+y^{2}}}&0\end{pmatrix}}.}
2.
d
d
x
[
sin
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x
)
+
C
]
=
d
d
x
sin
(
x
)
+
d
d
x
C
=
cos
(
x
)
+
0
=
cos
(
x
)
{\displaystyle 2.\qquad {\begin{aligned}{\frac {d}{dx}}[\sin(x)+C]&={\frac {d}{dx}}\sin(x)+{\frac {d}{dx}}C\\&=\cos(x)+0\\&=\cos(x)\end{aligned}}}
3.
⟨
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Δ
f
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f
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L
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∫
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∞
∞
f
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f
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x
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¯
d
x
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f
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f
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x
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¯
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∞
∞
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∫
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∞
∞
f
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f
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d
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∫
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∞
∞
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f
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x
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2
d
x
≥
0.
{\displaystyle 3.\qquad {\begin{aligned}\langle -\Delta f,f\rangle _{L^{2}}&=-\int _{-\infty }^{\infty }f''(x){\overline {f(x)}}\,dx\\[5pt]&=-\left[f'(x){\overline {f(x)}}\right]_{-\infty }^{\infty }+\int _{-\infty }^{\infty }f'(x){\overline {f'(x)}}\,dx\\[5pt]&=\int _{-\infty }^{\infty }\vert f'(x)\vert ^{2}\,dx\geq 0.\end{aligned}}}
4.
ln
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⋯
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{\displaystyle 4.\qquad {\begin{aligned}\ln(z)&={\frac {(z-1)^{1}}{1}}-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots \\&=\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {(z-1)^{k}}{k}}\end{aligned}}}
5.
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t
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x
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{\displaystyle 5.\qquad {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right),\\[2pt]x'&=\gamma \left(x-vt\right).\end{aligned}}}
6.
n
!
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{
n
⋅
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n
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!
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α
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if
n
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n
if
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and
(
n
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!
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α
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/
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n
+
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if
n
≤
0
and is not a negative multiple of
α
;
{\displaystyle 6.\qquad n!_{(\alpha )}={\begin{cases}n\cdot (n-\alpha )!_{(\alpha )}&{\text{ if }}n>\alpha \,;\\n&{\text{ if }}1\leq n\leq \alpha \,;{\text{and}}\\(n+\alpha )!_{(\alpha )}/(n+\alpha )&{\text{ if }}n\leq 0{\text{ and is not a negative multiple of }}\alpha \,;\end{cases}}}
7.
∫
0
∞
sin
t
t
d
t
=
lim
s
→
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∫
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∞
e
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t
sin
t
t
d
t
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lim
s
→
0
L
[
sin
t
t
]
=
lim
s
→
0
∫
s
∞
d
u
u
2
+
1
=
lim
s
→
0
arctan
u
|
s
∞
=
lim
s
→
0
[
π
2
−
arctan
(
s
)
]
=
π
2
.
{\displaystyle 7.\qquad {\begin{aligned}\int _{0}^{\infty }{\frac {\sin t}{t}}\,dt&=\lim _{s\to 0}\int _{0}^{\infty }e^{-st}{\frac {\sin t}{t}}\,dt=\lim _{s\to 0}{\mathcal {L}}\left[{\frac {\sin t}{t}}\right]\\[6pt]&=\lim _{s\to 0}\int _{s}^{\infty }{\frac {du}{u^{2}+1}}=\lim _{s\to 0}\arctan u{\Biggr |}_{s}^{\infty }\\[6pt]&=\lim _{s\to 0}\left[{\frac {\pi }{2}}-\arctan(s)\right]={\frac {\pi }{2}}.\end{aligned}}}
8.
∫
a
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x
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d
x
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a
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2
(
sec
θ
tan
θ
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ln
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sec
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tan
θ
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)
+
C
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2
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1
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ln
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ln
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{\displaystyle 8.\qquad {\begin{aligned}\int {\sqrt {a^{2}+x^{2}}}\,dx&={\frac {a^{2}}{2}}(\sec \theta \tan \theta +\ln |\sec \theta +\tan \theta |)+C\\[6pt]&={\frac {a^{2}}{2}}\left({\sqrt {1+{\frac {x^{2}}{a^{2}}}}}\cdot {\frac {x}{a}}+\ln \left|{\sqrt {1+{\frac {x^{2}}{a^{2}}}}}+{\frac {x}{a}}\right|\right)+C\\[6pt]&={\frac {1}{2}}\left(x{\sqrt {a^{2}+x^{2}}}+a^{2}\ln \left|{\frac {x+{\sqrt {a^{2}+x^{2}}}}{a}}\right|\right)+C.\end{aligned}}}
A: Eq 5, colon-indented w/out blank lines
5.
t
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γ
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t
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,
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{\displaystyle 5.\qquad {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right),\\[2pt]x'&=\gamma \left(x-vt\right).\end{aligned}}}
before or after.
B: Eq 5, colon-indented w/ blank line after, close before:
5.
t
′
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γ
(
t
−
v
x
c
2
)
,
x
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γ
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x
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t
)
.
{\displaystyle 5.\qquad {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right),\\[2pt]x'&=\gamma \left(x-vt\right).\end{aligned}}}
C: Eq 5, colon-indented w/ blank line before...
5.
t
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γ
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t
−
v
x
c
2
)
,
x
′
=
γ
(
x
−
v
t
)
.
{\displaystyle 5.\qquad {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right),\\[2pt]x'&=\gamma \left(x-vt\right).\end{aligned}}}
... and close after.
D: Eq 5, disp=block w/out blank lines
5.
t
′
=
γ
(
t
−
v
x
c
2
)
,
x
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=
γ
(
x
−
v
t
)
.
{\displaystyle 5.\qquad {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right),\\[2pt]x'&=\gamma \left(x-vt\right).\end{aligned}}}
before or after.
E: Eq 5, disp=block w/ blank line after, close before:
5.
t
′
=
γ
(
t
−
v
x
c
2
)
,
x
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=
γ
(
x
−
v
t
)
.
{\displaystyle 5.\qquad {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right),\\[2pt]x'&=\gamma \left(x-vt\right).\end{aligned}}}
F: Eq 5, disp=block w/ blank line before...
5.
t
′
=
γ
(
t
−
v
x
c
2
)
,
x
′
=
γ
(
x
−
v
t
)
.
{\displaystyle 5.\qquad {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right),\\[2pt]x'&=\gamma \left(x-vt\right).\end{aligned}}}
... and close after.
![{\displaystyle 2.\qquad {\begin{aligned}{\frac {d}{dx}}[\sin(x)+C]&={\frac {d}{dx}}\sin(x)+{\frac {d}{dx}}C\\&=\cos(x)+0\\&=\cos(x)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b9e86cf23794ff3e854b41e5edf2b71cec486b)
![{\displaystyle 3.\qquad {\begin{aligned}\langle -\Delta f,f\rangle _{L^{2}}&=-\int _{-\infty }^{\infty }f''(x){\overline {f(x)}}\,dx\\[5pt]&=-\left[f'(x){\overline {f(x)}}\right]_{-\infty }^{\infty }+\int _{-\infty }^{\infty }f'(x){\overline {f'(x)}}\,dx\\[5pt]&=\int _{-\infty }^{\infty }\vert f'(x)\vert ^{2}\,dx\geq 0.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e726cf83a1291402f58fcd9cb7b5e241a95fa572)
![{\displaystyle 5.\qquad {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right),\\[2pt]x'&=\gamma \left(x-vt\right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35c4c370a8f85c9972242ef0104d91cdc34d3671)
![{\displaystyle 7.\qquad {\begin{aligned}\int _{0}^{\infty }{\frac {\sin t}{t}}\,dt&=\lim _{s\to 0}\int _{0}^{\infty }e^{-st}{\frac {\sin t}{t}}\,dt=\lim _{s\to 0}{\mathcal {L}}\left[{\frac {\sin t}{t}}\right]\\[6pt]&=\lim _{s\to 0}\int _{s}^{\infty }{\frac {du}{u^{2}+1}}=\lim _{s\to 0}\arctan u{\Biggr |}_{s}^{\infty }\\[6pt]&=\lim _{s\to 0}\left[{\frac {\pi }{2}}-\arctan(s)\right]={\frac {\pi }{2}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70383eda9c445ccf63b6de737c8eced9e3497145)
![{\displaystyle 8.\qquad {\begin{aligned}\int {\sqrt {a^{2}+x^{2}}}\,dx&={\frac {a^{2}}{2}}(\sec \theta \tan \theta +\ln |\sec \theta +\tan \theta |)+C\\[6pt]&={\frac {a^{2}}{2}}\left({\sqrt {1+{\frac {x^{2}}{a^{2}}}}}\cdot {\frac {x}{a}}+\ln \left|{\sqrt {1+{\frac {x^{2}}{a^{2}}}}}+{\frac {x}{a}}\right|\right)+C\\[6pt]&={\frac {1}{2}}\left(x{\sqrt {a^{2}+x^{2}}}+a^{2}\ln \left|{\frac {x+{\sqrt {a^{2}+x^{2}}}}{a}}\right|\right)+C.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6048152954398cee0cca69de63d569ab8bf0898b)