Article 17
Exponential
Expressions.
Adding and subtracting (27), (28) give the identities
hence
The analogous exponential expressions for
are
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where the symbol stands
for the result of substituting
for in
the exponential development
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This will be more fully explained in treating of complex numbers, Arts. 28,
29.
-
Prob. 37.
- Show that the properties of the hyperbolic functions could be placed on a purely
algebraic basis by starting with equations (30) as their definitions; for example,
verify the identities:
-
Prob. 38.
- Prove .
-
Prob. 39.
- Assuming from Art. 14 that ,
satisfy the differential
equation , whose general
solution may be written ,
where ,
are arbitrary constants; show how to determine
,
in order to derive
the expressions for ,
,
respectively. [Use eq. (15).]
-
Prob. 40.
- Show how to construct a table of exponential functions from a table of hyperbolic
sines and cosines, and vice versa.
-
Prob. 41.
- Prove .
-
Prob. 42.
- Show that the area of any hyperbolic sector is infinite when its terminal line is one of
the asymptotes.
-
Prob. 43.
- From the relation
prove
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and examine the last term when
is odd or even. Find also the corresponding expression for
.