Hyperbolic functions occur frequently when describing conduction in bipolar devices, and it is helpful to list some of their important properties here. The hyperbolic functions are defined as follows:

B1

These functions can be easily visualized, as shown in Figures B.1 and B.2. From these figures we see that sinh, tanh, coth, and csch are odd functions, positive for positive arguments and negative for negative arguments. cosh and sech are even functions, positive for both positive and negative arguments. Thus we can write

B2

It is also useful to note that

B3

Hyperbolic functions of the sum of two arguments can be written

B4

The inverse hyperbolic functions can be written in terms of logarithms as

B5

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