# Elementary Transcentental Functions¶

These types of activation functions where highly explored by Taehwan Kim and Tulay Adali, mainly in [CIT2001-KIM] and [CIT2003-KIM]. Please refer to them for further information. These functions are divided into 4 groups.

• Circular
• Inverse Circular
• Hyperbolic
• Inverse Hyperbolic

## Circular¶

etf_circular_tan(z):

Computes tan of z element-wise.

$tan(z) = \frac{e^{iz} - e^{-iz}}{i(e^{iz} + e^{-iz})}$
Parameters: z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.
etf_circular_sin(z):

Computes sine of z element-wise.

$sin(z) = \frac{e^{iz} - e^{-iz}}{2i}$
Parameters: z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.

## Inverse Circular¶

etf_inv_circular_atan(z):

Computes the trignometric inverse tangent of z element-wise.

$atan(z) = \int_{0}^{z} \frac{dt}{1+t^2}$
Parameters: z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.
etf_inv_circular_asin(z):

Computes the trignometric inverse sine of z element-wise.

$asin(z) = \int_{0}^{z} \frac{dt}{(1-t)^1/2}$
Parameters: z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.
etf_inv_circular_acos(z):

Computes acos of z element-wise.

$acos(z) = \int_{z}^{1} \frac{dt}{(1-t^2)^1/2}$
Parameters: z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.

## Hyperbolic¶

etf_circular_tanh(z):

Computes hyperbolic tangent of z element-wise.

$tanh(z) = \frac{sinh(z)}{cosh(z)} = \frac{e^{z} - e^{-z}}{e^{z} + e^{-z}}$
Parameters: z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.
etf_circular_sinh(z):

Computes hyperbolic sine of z element-wise.

$sinh(z) = \frac{e^{z} - e^{-z}}{2}$
Parameters: z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.

## Inverse Hyperbolic¶

etf_inv_circular_atanh(z):

Computes inverse hyperbolic tangent of z element-wise.

$atanh(z) = \int_{0}^{z} \frac{dt}{1-t^2}$
Parameters: z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.
etf_inv_circular_asinh(z):

Computes inverse hyperbolic sine of z element-wise.

$asinh(z) = \int_{0}^{z} \frac{dt}{(1+t^2)^1/2}$
Parameters: z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.
 [CIT2001-KIM] Kim and T Adali “Complex Backpropagation neural network using elementary transdencental activation functions” 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221)
 [CIT2003-KIM] Kim and T Adali “Approximation by Fully Complex MLP Using Elementary Transcendental Activation Functions” 2001 Neural Computation