of ratios
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21—30 of 415 matching pages
21: 4.8 Identities
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4.8.2
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : principal branch of logarithm function, : phase and : complex variable
- A&S Ref:
- 4.1.7
- Permalink:
- http://dlmf.nist.gov/4.8.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.8(i), §4.8 and Ch.4
4.8.7
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : principal branch of logarithm function, : phase and : complex variable
- Referenced by:
- §4.2(i), §4.8(i)
- Permalink:
- http://dlmf.nist.gov/4.8.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.8(i), §4.8 and Ch.4
4.8.9
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : exponential function, : imaginary part, : principal branch of logarithm function and : complex variable
- A&S Ref:
- 4.2.3
- Permalink:
- http://dlmf.nist.gov/4.8.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.8(i), §4.8 and Ch.4
4.8.12
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : imaginary unit, : principal branch of logarithm function, : integer, : real or complex constant and : complex variable
- Permalink:
- http://dlmf.nist.gov/4.8.E12
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.8(i), §4.8 and Ch.4
4.8.15
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : phase, : real or complex constant and : complex variable
- A&S Ref:
- 4.2.16
- Permalink:
- http://dlmf.nist.gov/4.8.E15
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §4.8(ii), §4.8 and Ch.4
22: 7.2 Definitions
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7.2.1
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- Defines:
- : error function
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : integral and : complex variable
- A&S Ref:
- 7.1.1
- Referenced by:
- §7.5, §7.6(ii)
- Permalink:
- http://dlmf.nist.gov/7.2.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.2(i), §7.2 and Ch.7
7.2.2
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- Defines:
- : complementary error function
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : differential of , : error function, : base of natural logarithm, : integral and : complex variable
- A&S Ref:
- 7.1.2
- Referenced by:
- §7.5, §7.7(i)
- Permalink:
- http://dlmf.nist.gov/7.2.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.2(i), §7.2 and Ch.7
7.2.6
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- Defines:
- : Fresnel integral
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : differential of , : base of natural logarithm, : imaginary unit, : integral and : complex variable
- Referenced by:
- §7.5, §7.7(ii)
- Permalink:
- http://dlmf.nist.gov/7.2.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.2(iii), §7.2 and Ch.7
7.2.7
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- Defines:
- : Fresnel integral
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : cosine function, : differential of , : integral and : complex variable
- A&S Ref:
- 7.3.1
- Referenced by:
- §7.5, §7.6(i)
- Permalink:
- http://dlmf.nist.gov/7.2.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.2(iii), §7.2 and Ch.7
7.2.8
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- Defines:
- : Fresnel integral
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : sine function and : complex variable
- A&S Ref:
- 7.3.2
- Referenced by:
- §7.5, §7.6(i)
- Permalink:
- http://dlmf.nist.gov/7.2.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.2(iii), §7.2 and Ch.7
23: 7.6 Series Expansions
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7.6.1
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : error function, : factorial (as in ), : complex variable and : nonnegative integer
- A&S Ref:
- 7.1.5
- Referenced by:
- §7.6(i)
- Permalink:
- http://dlmf.nist.gov/7.6.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.6(i), §7.6 and Ch.7
7.6.2
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : error function, : base of natural logarithm, : complex variable and : nonnegative integer
- A&S Ref:
- 7.1.6
- Referenced by:
- §7.6(i), §7.6(ii)
- Permalink:
- http://dlmf.nist.gov/7.6.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.6(i), §7.6 and Ch.7
7.6.4
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ⓘ
- Symbols:
- : Fresnel integral, : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : complex variable and : nonnegative integer
- A&S Ref:
- 7.3.11
- Referenced by:
- §7.6(i)
- Permalink:
- http://dlmf.nist.gov/7.6.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.6(i), §7.6 and Ch.7
7.6.6
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ⓘ
- Symbols:
- : Fresnel integral, : the ratio of the circumference of a circle to its diameter, : factorial (as in ), : complex variable and : nonnegative integer
- A&S Ref:
- 7.3.13
- Referenced by:
- §7.6(i)
- Permalink:
- http://dlmf.nist.gov/7.6.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.6(i), §7.6 and Ch.7
7.6.10
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- Symbols:
- : Fresnel integral, : the ratio of the circumference of a circle to its diameter, : spherical Bessel function of the first kind, : complex variable and : nonnegative integer
- Referenced by:
- §7.6(ii)
- Permalink:
- http://dlmf.nist.gov/7.6.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.6(ii), §7.6 and Ch.7
24: 25.4 Reflection Formulas
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25.4.1
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- Symbols:
- : gamma function, : Riemann zeta function, : the ratio of the circumference of a circle to its diameter, : cosine function and : complex variable
- Keywords:
- reflection formula
- Source:
- Apostol (1976, (12), p. 259)
- Referenced by:
- §25.10(i), §5.17
- Permalink:
- http://dlmf.nist.gov/25.4.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.4 and Ch.25
25.4.2
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ⓘ
- Symbols:
- : gamma function, : Riemann zeta function, : the ratio of the circumference of a circle to its diameter, : sine function and : complex variable
- Keywords:
- reflection formula
- Source:
- Apostol (1976, (13), p. 259)
- A&S Ref:
- 23.2.6
- Permalink:
- http://dlmf.nist.gov/25.4.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.4 and Ch.25
25.4.4
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- Defines:
- : Riemann’s -function
- Symbols:
- : gamma function, : Riemann zeta function, : the ratio of the circumference of a circle to its diameter and : complex variable
- Keywords:
- definition
- Source:
- Apostol (1976, p. 260)
- Permalink:
- http://dlmf.nist.gov/25.4.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.4 and Ch.25
25.4.5
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ⓘ
- Symbols:
- : gamma function, : Riemann zeta function, : binomial coefficient, : the ratio of the circumference of a circle to its diameter, : cosine function, : imaginary part, : real part, : sine function, : nonnegative integer, : nonnegative integer, : complex variable and
- Keywords:
- derivative
- Source:
- Apostol (1985a, (6), p. 224)
- Permalink:
- http://dlmf.nist.gov/25.4.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.4 and Ch.25
25.4.6
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- Symbols:
- : the ratio of the circumference of a circle to its diameter, : equals by definition, : imaginary unit, : principal branch of logarithm function and
- Keywords:
- definition
- Source:
- Apostol (1985a, p. 223)
- Referenced by:
- §25.6(ii)
- Permalink:
- http://dlmf.nist.gov/25.4.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §25.4 and Ch.25
25: 7.5 Interrelations
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7.5.1
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ⓘ
- Symbols:
- : Dawson’s integral, : the ratio of the circumference of a circle to its diameter, : error function, : Faddeeva (or Faddeyeva) function, : base of natural logarithm, : imaginary unit and : complex variable
- Referenced by:
- §7.5
- Permalink:
- http://dlmf.nist.gov/7.5.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.5 and Ch.7
7.5.3
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ⓘ
- Symbols:
- : auxiliary function for Fresnel integrals, : auxiliary function for Fresnel integrals, : Fresnel integral, : the ratio of the circumference of a circle to its diameter, : cosine function, : sine function and : complex variable
- A&S Ref:
- 7.3.9
- Referenced by:
- §7.12(ii), §7.5
- Permalink:
- http://dlmf.nist.gov/7.5.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.5 and Ch.7
7.5.5
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ⓘ
- Symbols:
- : auxiliary function for Fresnel integrals, : auxiliary function for Fresnel integrals, : Fresnel integral, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit and : complex variable
- Referenced by:
- §7.13(iv), §7.5, §7.7(ii)
- Permalink:
- http://dlmf.nist.gov/7.5.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.5 and Ch.7
7.5.7
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- Defines:
- : change of variable (locally)
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : imaginary unit and : complex variable
- Referenced by:
- §7.12(ii), §7.5
- Permalink:
- http://dlmf.nist.gov/7.5.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.5 and Ch.7
7.5.13
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ⓘ
- Symbols:
- : Dawson’s integral, : Goodwin–Staton integral, : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : exponential integral and : real variable
- Referenced by:
- §7.5
- Permalink:
- http://dlmf.nist.gov/7.5.E13
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §7.5 and Ch.7
26: 11.4 Basic Properties
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11.4.5
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- Symbols:
- : Struve function, : the ratio of the circumference of a circle to its diameter, : cosine function and : complex variable
- A&S Ref:
- 12.1.16
- Referenced by:
- §11.4(i)
- Permalink:
- http://dlmf.nist.gov/11.4.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §11.4(i), §11.4 and Ch.11
11.4.6
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ⓘ
- Symbols:
- : Struve function, : the ratio of the circumference of a circle to its diameter, : sine function and : complex variable
- Permalink:
- http://dlmf.nist.gov/11.4.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §11.4(i), §11.4 and Ch.11
11.4.7
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : hyperbolic cosine function, : modified Struve function and : complex variable
- Permalink:
- http://dlmf.nist.gov/11.4.E7
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §11.4(i), §11.4 and Ch.11
11.4.8
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : hyperbolic sine function, : modified Struve function and : complex variable
- Permalink:
- http://dlmf.nist.gov/11.4.E8
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §11.4(i), §11.4 and Ch.11
11.4.10
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- Symbols:
- : Struve function, : the ratio of the circumference of a circle to its diameter, : cosine function, : sine function and : complex variable
- Permalink:
- http://dlmf.nist.gov/11.4.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §11.4(i), §11.4 and Ch.11
27: 22.11 Fourier and Hyperbolic Series
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22.11.1
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- Symbols:
- : Jacobian elliptic function, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : nome, : sine function, : complex, : modulus and : change of variable
- A&S Ref:
- 16.23.1
- Permalink:
- http://dlmf.nist.gov/22.11.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.11 and Ch.22
22.11.2
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ⓘ
- Symbols:
- : Jacobian elliptic function, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : cosine function, : nome, : complex, : modulus and : change of variable
- A&S Ref:
- 16.23.2
- Permalink:
- http://dlmf.nist.gov/22.11.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.11 and Ch.22
22.11.3
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ⓘ
- Symbols:
- : Jacobian elliptic function, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : cosine function, : nome, : complex, : modulus and : change of variable
- A&S Ref:
- 16.23.3
- Permalink:
- http://dlmf.nist.gov/22.11.E3
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.11 and Ch.22
22.11.4
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ⓘ
- Symbols:
- : Jacobian elliptic function, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : cosine function, : nome, : complex, : modulus and : change of variable
- A&S Ref:
- 16.23.4
- Permalink:
- http://dlmf.nist.gov/22.11.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.11 and Ch.22
22.11.5
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ⓘ
- Symbols:
- : Jacobian elliptic function, : the ratio of the circumference of a circle to its diameter, : Legendre’s complete elliptic integral of the first kind, : nome, : sine function, : complex, : modulus, : complementary modulus and : change of variable
- A&S Ref:
- 16.23.5
- Permalink:
- http://dlmf.nist.gov/22.11.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §22.11 and Ch.22
28: 10.34 Analytic Continuation
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10.34.1
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : modified Bessel function of the first kind, : integer, : complex variable and : complex parameter
- A&S Ref:
- 9.6.30
- Referenced by:
- §10.30(ii), §10.34, §10.47(v)
- Permalink:
- http://dlmf.nist.gov/10.34.E1
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.34 and Ch.10
10.34.2
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : cosecant function, : base of natural logarithm, : imaginary unit, : modified Bessel function of the first kind, : modified Bessel function of the second kind, : sine function, : integer, : complex variable and : complex parameter
- A&S Ref:
- 9.6.31
- Referenced by:
- §10.27, §10.34, §10.34
- Permalink:
- http://dlmf.nist.gov/10.34.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.34 and Ch.10
10.34.4
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : cosecant function, : base of natural logarithm, : imaginary unit, : modified Bessel function of the second kind, : sine function, : integer, : complex variable and : complex parameter
- Referenced by:
- §10.34, §10.40(i)
- Permalink:
- http://dlmf.nist.gov/10.34.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.34 and Ch.10
10.34.5
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : modified Bessel function of the first kind, : modified Bessel function of the second kind, : integer, : integer and : complex variable
- Permalink:
- http://dlmf.nist.gov/10.34.E5
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.34 and Ch.10
10.34.6
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ⓘ
- Symbols:
- : the ratio of the circumference of a circle to its diameter, : base of natural logarithm, : imaginary unit, : modified Bessel function of the second kind, : integer, : integer and : complex variable
- Referenced by:
- §10.40(i)
- Permalink:
- http://dlmf.nist.gov/10.34.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §10.34 and Ch.10
29: 11.7 Integrals and Sums
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11.7.2
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ⓘ
- Symbols:
- : gamma function, : Struve function, : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : complex variable and : real or complex order
- A&S Ref:
- 12.1.24
- Permalink:
- http://dlmf.nist.gov/11.7.E2
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §11.7(i), §11.7 and Ch.11
11.7.4
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ⓘ
- Symbols:
- : gamma function, : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : modified Struve function, : complex variable and : real or complex order
- Permalink:
- http://dlmf.nist.gov/11.7.E4
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §11.7(i), §11.7 and Ch.11
11.7.6
.
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ⓘ
- Symbols:
- : gamma function, : Struve function, : the ratio of the circumference of a circle to its diameter, : real part, : complex variable and : real or complex order
- A&S Ref:
- 12.1.28 (The condition on has been corrected.)
- Permalink:
- http://dlmf.nist.gov/11.7.E6
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §11.7(i), §11.7 and Ch.11
11.7.9
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ⓘ
- Symbols:
- : Struve function, : the ratio of the circumference of a circle to its diameter, : cotangent function, : differential of , : integral, : real part and : real or complex order
- Permalink:
- http://dlmf.nist.gov/11.7.E9
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §11.7(ii), §11.7 and Ch.11
11.7.10
,
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ⓘ
- Symbols:
- : gamma function, : Struve function, : the ratio of the circumference of a circle to its diameter, : differential of , : integral, : real part and : real or complex order
- Keywords:
- Mellin transform
- Permalink:
- http://dlmf.nist.gov/11.7.E10
- Encodings:
- pMML, png, TeX
- See also:
- Annotations for §11.7(ii), §11.7 and Ch.11
30: 20.3 Graphics
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