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# Mathematical Constants
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High-precision mathematical constants including π, e, and various special constants from number theory and analysis. All constants are computed with arbitrary precision matching the current context precision.
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## Capabilities
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### Fundamental Constants
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Basic mathematical constants that appear throughout mathematics.
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```python { .api }
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# Fundamental constants (accessed as mp.constant_name)
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mp.pi        # π ≈ 3.14159265358979323846...
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mp.e         # Euler's number e ≈ 2.71828182845904523536...
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mp.ln2       # Natural logarithm of 2 ≈ 0.69314718055994530942...
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mp.ln10      # Natural logarithm of 10 ≈ 2.30258509299404568402...
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mp.phi       # Golden ratio φ = (1+√5)/2 ≈ 1.61803398874989484820...
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```
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### Special Mathematical Constants
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Important constants from number theory and analysis.
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```python { .api }
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# Number theory and analysis constants
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mp.euler     # Euler-Mascheroni constant γ ≈ 0.57721566490153286061...
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mp.catalan   # Catalan's constant G ≈ 0.91596559417721901505...
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mp.khinchin  # Khinchin's constant K ≈ 2.68545200106530644531...
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mp.glaisher  # Glaisher-Kinkelin constant A ≈ 1.28242712910062263688...
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mp.apery     # Apéry's constant ζ(3) ≈ 1.20205690315959428540...
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```
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### Prime-Related Constants
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Constants related to prime numbers and number theory.
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```python { .api }
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# Prime number constants  
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mp.twinprime # Twin prime constant ≈ 0.66016181584686957392...
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mp.mertens   # Mertens constant M ≈ 0.26149721284764278375...
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```
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### Angular and Unit Conversion
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Constants for angle conversion and unit systems.
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```python { .api }
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# Unit conversion
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mp.degree    # π/180 for degree to radian conversion ≈ 0.01745329251994329577...
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```
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### Special Values
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Special numerical values used in computations.
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```python { .api }
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# Special values
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mp.inf       # Positive infinity
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mp.ninf      # Negative infinity (-∞)
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mp.nan       # Not a number (NaN)
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mp.j         # Imaginary unit i = √(-1)
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mp.eps       # Machine epsilon (smallest representable positive number)
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```
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### Accessing Constants
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All constants automatically adjust to the current precision setting and can be accessed as attributes of the mpmath context.
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```python { .api }
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# Constants adapt to current precision
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with mp.workdps(50):
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    high_precision_pi = mp.pi  # π with 50 decimal places
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with mp.workdps(100):  
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    very_high_precision_e = mp.e  # e with 100 decimal places
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```
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### Constant Relationships
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Mathematical relationships between constants that can be verified with high precision.
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## Usage Examples
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```python
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import mpmath
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from mpmath import mp
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# Set different precision levels
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mp.dps = 30
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# Display fundamental constants
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print("Fundamental Constants:")
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print(f"π = {mp.pi}")
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print(f"e = {mp.e}")
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print(f"Golden ratio φ = {mp.phi}")
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print(f"ln(2) = {mp.ln2}")
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print(f"ln(10) = {mp.ln10}")
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# Special constants
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print("\nSpecial Constants:")
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print(f"Euler-Mascheroni γ = {mp.euler}")
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print(f"Catalan's constant = {mp.catalan}")
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print(f"Khinchin's constant = {mp.khinchin}")
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print(f"Glaisher-Kinkelin A = {mp.glaisher}")
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print(f"Apéry's constant ζ(3) = {mp.apery}")
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# Prime-related constants
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print("\nPrime-Related Constants:")
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print(f"Twin prime constant = {mp.twinprime}")
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print(f"Mertens constant = {mp.mertens}")
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# Unit conversion
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print("\nUnit Conversion:")
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print(f"Degrees to radians: π/180 = {mp.degree}")
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angle_deg = 90
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angle_rad = angle_deg * mp.degree
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print(f"{angle_deg}° = {angle_rad} radians")
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# Verify relationships
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print("\nConstant Relationships:")
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print(f"φ² - φ - 1 = {mp.phi**2 - mp.phi - 1}")  # Should be 0
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print(f"e^(iπ) + 1 = {mp.exp(mp.j * mp.pi) + 1}")  # Euler's identity, should be 0
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print(f"ζ(2) = π²/6 = {mp.pi**2 / 6}")
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print(f"Actual ζ(2) = {mp.zeta(2)}")
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# Precision scaling
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print("\nPrecision Scaling:")
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with mp.workdps(10):
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    pi_10 = mp.pi
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    print(f"π with 10 digits: {pi_10}")
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with mp.workdps(50):
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    pi_50 = mp.pi  
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    print(f"π with 50 digits: {pi_50}")
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with mp.workdps(100):
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    pi_100 = mp.pi
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    print(f"π with 100 digits: {pi_100}")
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# Special values
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print("\nSpecial Values:")
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print(f"Positive infinity: {mp.inf}")
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print(f"Negative infinity: {mp.ninf}")
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print(f"Not a number: {mp.nan}")
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print(f"Imaginary unit: {mp.j}")
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print(f"j² = {mp.j**2}")  # Should be -1
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# Machine epsilon  
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print(f"Machine epsilon: {mp.eps}")
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# Using constants in calculations
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print("\nCalculations with Constants:")
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# Area of unit circle
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area = mp.pi * 1**2
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print(f"Area of unit circle: π = {area}")
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# Compound interest formula: A = P*e^(rt)
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principal = 1000
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rate = 0.05  # 5%
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time = 10
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amount = principal * mp.exp(rate * time)
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print(f"Compound interest: {principal} * e^({rate}*{time}) = {amount}")
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# Golden ratio properties
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print(f"φ = (1 + √5)/2 = {(1 + mp.sqrt(5))/2}")
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print(f"1/φ = φ - 1 = {1/mp.phi} = {mp.phi - 1}")
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# Trigonometric identities with high precision
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print(f"sin²(π/6) + cos²(π/6) = {mp.sin(mp.pi/6)**2 + mp.cos(mp.pi/6)**2}")
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print(f"e^(iπ/4) = {mp.exp(mp.j * mp.pi/4)}")
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print(f"cos(π/4) + i*sin(π/4) = {mp.cos(mp.pi/4) + mp.j*mp.sin(mp.pi/4)}")
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# Natural logarithm identities
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print(f"e^(ln(2)) = {mp.exp(mp.ln2)}")  # Should be 2
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print(f"ln(e) = {mp.ln(mp.e)}")        # Should be 1
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print(f"ln(10) = {mp.ln10}")
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print(f"ln(10)/ln(2) = {mp.ln10/mp.ln2}")  # log₂(10)
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```
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### Historical and Mathematical Context
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These constants have deep mathematical significance:
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- **π (pi)**: Ratio of circumference to diameter of a circle, appears throughout geometry and analysis
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- **e**: Base of natural logarithm, fundamental in calculus and probability
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- **γ (Euler-Mascheroni)**: Appears in number theory and analysis, related to harmonic numbers
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- **φ (Golden ratio)**: Appears in geometry, art, and nature; satisfies φ² = φ + 1
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- **G (Catalan)**: Arises in combinatorics and number theory
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- **K (Khinchin)**: Related to continued fraction expansions
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- **A (Glaisher-Kinkelin)**: Appears in various mathematical series and products
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- **ζ(3) (Apéry)**: Value of Riemann zeta function at 3, proved irrational by Apéry
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All constants are computed using mpmath's arbitrary precision arithmetic, ensuring accuracy to any requested precision level. The library uses efficient algorithms and series expansions optimized for high-precision computation.