Mary Nanfuka

1. **Inverse of Mittag-Leffler Function**:

The inverse of the Mittag-Leffler function is not easily obtainable in a closed-form expression. However, it can be approximated numerically using various methods, such as:

- Numerical root-finding algorithms (e.g., Newton-Raphson method)

- Series expansion techniques

- Numerical inversion of the Laplace transform

2. **Partial Derivative of Mittag-Leffler Function**:

The partial derivative of the Mittag-Leffler function with respect to its parameters can be computed using the following formula:

```

∂Eα,β(z) / ∂α = -z * Eα+1,β+1(z)

∂Eα,β(z) / ∂β = -Eα+1,β+1(z)

```

where Eα,β(z) is the Mittag-Leffler function defined as:

```

Eα,β(z) = Σ(k=0 to ∞) z^k / Γ(αk + β)

```

Here, Γ(x) represents the Gamma function.

3. **Product of Mittag-Leffler Functions**:

The product of two Mittag-Leffler functions can be computed using the following formula:

```

Eα1,β1(z) * Eα2,β2(z) = Eα1+α2,β1+β2(z)

```

This formula allows you to express the product of two Mittag-Leffler functions as a single Mittag-Leffler function with the sum of the respective parameters.

To implement these computations in code, you can use programming languages such as Python, MATLAB, or R, and leverage existing libraries or functions that provide support for the Mittag-Leffler function, such as:

- **Python**: The `scipy.special.mittag_leffler` function from the SciPy library can be used to evaluate the Mittag-Leffler function.

- **MATLAB**: The `mlf` function from the MATLAB Central File Exchange can be used to compute the Mittag-Leffler function.

- **R**: The `mittag.leffler` function from the `pracma` package can be used to evaluate the Mittag-Leffler function.

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