Appendix C: Mathematical Formulas
C.1 Formulas Typically Used in Perturbation Analysis
In particular perturbation analysis involves using a lot of mathematical standard
operations involving complex variables, powers of two-term sums, and averaging
integrals. Here are formulas for most of them. Unless otherwise stated all variables
are real-valued, i denotes the imaginary unit, overbars complex conjugation, and cc
complex conjugates of preceding terms.
C.1.1 Complex Numbers
In the below formulas A, B, and i are complex-valued, while all other variables are
real-valued.
i ¼ Ài;
a þ ib ¼ a À ib;
AB ¼
A
B; A; B 2 C;
A n ¼
A
n ; A 2 C;
A ¼ ae
iu ; where a ¼ A
j j ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ReðAÞ
2 þ ImðAÞ
2
q
;
and u ¼ \A ¼ arctan ImðAÞ=ReðAÞ
ð
Þ ;
e ia ¼ e
Àia ;
e
ia ¼ cos a þ i sin a;
cos u ¼
1
2
ðe
iu þ e
Àiu Þ; sin u ¼
1
2i
ðe
iu À e
Àiu Þ;
ffiffiffiffiffiffiffiffiffiffiffi ffi
a þ ib
p
¼ a þ ib; where; ab ¼
1
2
b; a
2 À b
2 ¼ a;
and a ¼ Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1
2
a þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a 2 þ b 2
p
r
; b ¼ Æ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
2
Àa þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a 2 þ b 2
p
r
:
ðC:1Þ
© The Editor(s) (if applicable) and The Author(s), under exclusive license to
Springer Nature Switzerland AG 2021
J. J. Thomsen Vibrations and Stability,
https://doi.org/10.1007/978-3-030-68045-9
477