Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS
A rational function is of the form f …x† ˆp n …x†=q n …x†, where p n …x† and q n …x†
are polynomials.
A transcendental function is any function which is not algebraic, for example,
the trigonometric functions sin x, cos x, etc., the exponential functions e x , the
logarithmic functions log x, and the hyperbolic functions sinh x, cosh x, etc.
The exponential functions obey the index law. The logarithmic functions are
inverses of the exponential functions, that is, if a x ˆ y then x ˆ log a y, where a is
called the base of the logarithm. If a ˆ e, which is often called the natural base of
logarithms, we denote log e x by ln x, called the natural logarithm of x. The fundamental
rules obeyed by logarithms are
ln…mn† ˆln m ‡ ln n; ln…m=n† ˆln m ln n; and ln m p ˆ p ln m:
The hyperbolic functions are de®ned in terms of exponential functions as
follows
sinh x ˆ ex e x
; cosh x ˆ ex ‡ e x
;
2
2
tanh x ˆ sinh x
cosh x ˆ ex e x
e x ‡ e x ; coth x ˆ 1
tanh x ˆ ex ‡ e x
e x e x ;
sech x ˆ 1
cosh x ˆ
2
1
e x ‡ ex ; cosech x ˆ
sinh x ˆ
2
e x e x :
Rough graphs of these six functions are given in Fig. A1.2.
Some fundamental relationships among these functions are as follows:
cosh 2 x sinh 2 x ˆ 1; sech 2 x ‡ tanh 2 x ˆ 1; coth 2 x cosech 2 x ˆ 1;
sinh…x y† ˆsinh x cosh y cosh x sinh y;
cosh…x y† ˆcosh x cosh y sinh x sinh y;
tanh…x y† ˆ
tanh x tanh y
1 tanh x tanh y :
Figure A1.2.
Hyperbolic functions.
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