The sigmoid function and the hyperbolic tangent (tanh) function are both activation functions that are commonly used in neural networks.

The sigmoid function takes any real-valued input and maps it to the range 0 to 1, outputting a probability-like value. It is defined as:

sigmoid(x) = 1 / (1 + e^-x)

The tanh function, on the other hand, maps real-valued input to the range -1 to 1. It is defined as:

tanh(x) = 2 * sigmoid(2x) - 1

Both the sigmoid and tanh functions are widely used in neural networks, and they have some similar properties:

- Both functions are non-linear, which makes them useful for capturing complex patterns in data.

- Both functions are smooth, which makes them easier to optimize than other non-linear functions such as the ReLU function.

- Both functions are monotonic, which means that they increase or decrease monotonically with respect to the input.

However, there are also some differences between the sigmoid and tanh functions:

- The range of the sigmoid function is fixed at 0 to 1, whereas the range of the tanh function is fixed at -1 to 1.

- The sigmoid function has a slower rate of change than the tanh function, which means that the derivative of the sigmoid function will be smaller than the derivative of the tanh function for a given input. This can affect the speed at which a neural network trained with the sigmoid function will converge compared to a network trained with the tanh function.

- The sigmoid function is often used in the output layer of a binary classification network, while the tanh function is often used in the hidden layers of a network.