Exponential Integral Calculator
Calculate the exponential integral Ei(x) and related functions with step-by-step solutions
Ei(x) = γ + ln|x| + Σ(xⁿ / n·n!)
Where γ ≈ 0.5772 is the Euler-Mascheroni constant | Ei(1) ≈ 1.8951178163
Note: Ei(x) has a singularity at x = 0 E₁(x) = ∫ₓ^∞ (e⁻ᵗ/t) dt, defined for x > 0 Eₙ(x) = ∫₁^∞ (e⁻ˣᵗ/tⁿ) dt, generalized form
Quick Examples
Error
Ei() E₁() E()
Derivative -Eₙ₋₁(x)
E₁(x)
Input Value
Step-by-Step Solution
Identify the input value
x =
Apply the Exponential Integral Ei(x) Apply E₁(x) = ∫ₓ^∞ (e⁻ᵗ/t) dt Apply Eₙ(x) = ∫₁^∞ (e⁻ˣᵗ/tⁿ) dt
Using series: Ei(x) = γ + ln|x| + Σ(xⁿ/n·n!) Using continued fraction for numerical stability Using recurrence: Eₙ₊₁(x) = (1/n)[e⁻ˣ - x·Eₙ(x)]
Result
Ei() = E₁() = E() =
Special Values Reference
| x | Ei(x) | E₁(x) |
|---|---|---|
| 0 | -∞ (singularity) | +∞ |
| 0.5 | ≈ 0.4542 | ≈ 0.5598 |
| 1 | ≈ 1.8951 | ≈ 0.2194 |
| 2 | ≈ 4.9542 | ≈ 0.0489 |
| -1 | ≈ -0.2194 | N/A (x > 0 required) |
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About Exponential Integral Calculator
What is the Exponential Integral?
The Exponential Integral is a special function in mathematics that appears frequently in physics, engineering, and applied mathematics. It is defined as an integral involving the exponential function divided by its argument.
Mathematical Definition
Principal Exponential Integral Ei(x)
For real non-zero values of x, the exponential integral Ei(x) is defined as:
Ei(x) = -∫₋ₓ^∞ (e⁻ᵗ/t) dt = ∫₋∞^x (eᵗ/t) dt
This integral is understood in terms of the Cauchy principal value due to the singularity at t = 0.
Series Expansion
For computational purposes, Ei(x) can be calculated using the series expansion:
Ei(x) = γ + ln|x| + x + x²/(2·2!) + x³/(3·3!) + x⁴/(4·4!) + ...
where γ ≈ 0.5772156649 is the Euler-Mascheroni constant.
Related Functions
Generalized Exponential Integral Eₙ(x)
For n = 1, 2, 3, ... and x > 0:
Eₙ(x) = ∫₁^∞ (e⁻ˣᵗ/tⁿ) dt
Relationship Between Ei and E₁
For x > 0: Ei(x) = -E₁(-x)
Key Properties
Derivative
d/dx [Ei(x)] = eˣ/x
Recurrence Relation
Eₙ₊₁(x) = (1/n)[e⁻ˣ - x·Eₙ(x)]
Special Values
- Ei(1) ≈ 1.8951178163
- E₁(1) ≈ 0.2193839345
- Ei(-1) ≈ -0.2193839345
Singularity
Ei(x) has a logarithmic singularity at x = 0.
Applications
- Physics - Heat conduction, radiation transport, quantum mechanics
- Engineering - Signal processing, electromagnetic theory
- Probability - Related to the logarithmic integral li(x)
- Number Theory - Prime counting function approximations
- Astrophysics - Stellar atmosphere calculations
Common Mistakes
- Sign conventions - Different sources use different definitions; be aware of the sign in the integral
- Confusing Ei and E₁ - These are related but different functions
- Domain issues - Ei(x) is undefined at x = 0 (has a singularity)