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Complex numbers •Definitions •Conversions •Arithmetic •Hyperbolic Functions Define the imaginary number i 1 so that i 2 1, If z x iy then x is the real part of z and y is the imaginary part i 3 i i 2 i i 4 i 2i 2 1 1 1 Im z Argand diagram x iy y r x Main page Re z Complex numbers: Definitions If z x iy then the conjugate of z , written z or z * is x iy If the complex number z x iy then the Modulus of z is written as z and the Argument of z is written as Arg (z ) so that z x 2 y 2 r , Arg ( z ) tan 1 ( y / x) r , are shown in the Argand diagram Cartesian form (Real/Imaginary form) z x iy Polar form Complex numbers: Forms If is the principal argument of a complex number z then Re z Main page Re z Polar to Cartesian form x r cos y r sin Cartesian to Polar form Eulers formula cos i sin ei Im z r x Principal argument x iy y (Modulus/Argument form) z r (cos i sin ) r cis Im z Exponential form z re i r x2 y2 tan 1 ( y / x) NB. You may need to add or 1 subtract to tan ( y / x) in order that gives z in the correct quadrant Let z a ib and w c id Multiplication z w (a ib )(c id ) (ac bd ) i (ad bc) Addition/ subtraction z w (a c) i (b d ) Equivalence Complex numbers: Arithmetic z w a c and b d De Moivres theorem Division (cos i sin ) n cos n i sin n z z w z w (a ib )(c id ) w w w w 2 c2 d 2 Polar/ exponential form: Powers/ roots ac bd bc ad 2 i 2 2 2 c d c d If z then r n ei ( n ) Polar/ exponential form: Mult/division If z then r cis re i and w s cis se i z w rs cis( ) rse and z r r cis( ) e i( ) w s s r cis re i z n r n cis (n ) and n n z n r cis n i( ) re n i Main page Hyperbolic Sine & Cosine Functions cosh x 1 x 1 x e e x and sinh x e e x 2 2 Other Hyperbolic Functions sinh x tanh x , cosh x 1 cosechx , sinh x 1 sechx cosh x cosh x 1 coth x sinh x tanh x Equivalences Main page cos i sin ei cos i cosh and sin i i sinh cosh i cos and sinh i i sin Complex numbers: Hyperbolic Functions Sine & Cosine Functions in Exponential form 1 i e e i and 2 1 i sin e e i 2i cos Eulers formula Complex numbers That’s all folks! Main page