Maths |
Complex Numbers |
| Complex Numbers |
| Note: Some textbooks use the letter j to represent the imaginary part of a complex number. I have used the more universal i throughout. |
| A complex number, z, is of the form: |
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z = x + iy |
| or, using polar coordinates : |
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z = [ r , θ ] |
| where x and y are real numbers and: |
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i = - 1 |
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i 2 = - 1 |
| The modulus of a complex number is: |
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| z | = x 2 + y 2 |
| The argument of a complex number is: |
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arg ( z ) = tan - 1 ( y x ) |
| The conjugate of a complex number is: |
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z * = x − iy |
| In the following lines: |
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z 1 = a + ib |
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z 2 = c + id |
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z 1 + z 2 = ( a + bi ) + ( c + di ) = ( a + c ) + ( b + d ) i |
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z 1 − z 2 = ( a + bi ) − ( c + di ) = ( a − c ) + ( b − d ) i |
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z 1 z 2 = ac + adi + bci + bdi 2 = ( ac − bd ) + ( ad + bc ) i |
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z 1 z 2 = a + bi c + di = ( a + bi ) ( c - di ) ( c + di ) ( c - di ) = ac - adi + bci - bdi 2 c 2 - d 2 i 2 = ac + bd + ( bc - ad ) i c 2 + d 2 = ac + bd c 2 + d 2 + ( bc - ad c 2 + d 2 ) i |
| For z = x + iy and where y is measured in radians: |
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e z = e x ( cos y + i sin y ) |
When θ is measured in radians:
The exponential form of a complex number: |
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[ r , θ ] = r e iθ |
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e x + iy = e x ( cos y + i sin y ) |
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a z = e zlna , a ∈ ℝ + |
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e z 1 × e z 2 = e z 1 + z 2 |
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e z 1 e z 2 = e z 1 − z 2 |
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ln ( r e iθ ) = ln r + i ( θ + 2 n θ ) , n ∈ ℤ |
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a x + iy = a x ( cos ( y ln a ) + i sin ( y ln a ) ) |
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e iz = cos z + i sin z |
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sin z = e iz − e − iz 2 i |
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cos z = e iz + e − iz 2 |
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sec z = 1 cos z = 2 e iz + e − iz |
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csc z = 1 sin z = 2 i e iz − e − iz |
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tan z = sin z coz z = e iz − e − iz i ( e iz + e − iz ) |
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cot z = cos z sin z = i ( e iz + e − iz ) e iz − e − iz |
| Complex Trigonometric Function Properties: |
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sin 2 z + cos 2 z = 1 |
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1 + tan 2 z = sec 2 z |
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1 + cot 2 z = csc 2 z |
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sin ( - z ) = - sin z |
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cos ( - z ) = cos z |
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tan ( - z ) = - tan z |
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sin ( z 1 ± z 2 ) = sin z 1 cos z 2 ± cos z 1 sin z 2 |
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cos ( z 1 ± z 2 ) = cos z 1 co s z 2 ∓ sin z 1 sin z 2 |
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tan ( z 1 ± z 2 ) = tan z 1 ± tan z 2 1 ∓ tan z 1 tan z 2 |
| Complex Hyperbolic Functions: |
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sinh z = e z - e - z 2 |
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cosh z = e z + e - z 2 |
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sech z = 1 cosh z = 2 e z + e - z |
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csch z = 1 sinh z = 2 e z - e - z |
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tanh z = sinh z cosh z = e z - e - z e z + e - z |
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coth z = cosh z sinh z = e z + e - z e z - e - z |
| Complex Hyperbolic Properties: |
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cosh 2 z − sinh 2 z = 1 |
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1 − tanh 2 z = sech 2 z |
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coth 2 z − 1 = csch 2 z |
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sinh ( − z ) = − sinh z |
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cosh ( − z ) = cosh z |
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tanh ( − z ) = − tanh z |
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sinh ( z 1 ± z 2 ) = sinh z 1 cosh z 2 ± cosh z 1 sinh z 2 |
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cosh ( z 1 ± z 2 ) = cosh z 1 cosh z 2 ± sinh z 1 sinh z 2 |
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tanh ( z 1 ± z 2 ) = tanh z 1 ± tanh z 2 1 ± tanh z 1 tanh z 2 |
| Complex Trigonometric and Hyperbolic Function Relations: |
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sin iz = i sinh z |
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cos iz = cosh z |
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tan iz = i tanh z |
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sinh iz = i sin z |
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cosh iz = cos z |
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tanh iz = i tan z |
| Complex Inverse Trigonometric Functions: |
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sin − 1 z = 1 i ln ( iz + 1 − z 2 + 2 k π i ) , k ∈ ℤ |
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cos − 1 z = 1 i ln ( z + z 2 - 1 + 2 k π i ) , k ∈ ℤ |
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csc − 1 z = 1 i ln ( i + z 2 - 1 z + 2 k π i ) , k ∈ ℤ |
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sec − 1 z = 1 i ln ( 1 + 1 − z 2 z + 2 k π i ) , k ∈ ℤ |
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tan − 1 z = 1 2 i ln ( 1 + iz 1 − iz + 2 k π i ) , k ∈ ℤ |
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cot − 1 z = 1 2 i ln ( z + i z − i + 2 k π i ) , k ∈ ℤ |
| Complex Inverse Hyperbolic Functions: |
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sinh − 1 z = ln ( z + z 2 + 1 + 2 k π i ) , k ∈ ℤ |
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cosh − 1 z = ln ( z + z 2 − 1 + 2 k π i ) , k ∈ ℤ |
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csch − 1 z = ln ( 1 + z 2 + 1 z + 2 k π i ) , k ∈ ℤ |
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sech − 1 z = ln ( 1 + 1 − z 2 z + 2 k π i ) , k ∈ ℤ |
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tanh − 1 z = 1 2 ln ( 1 + z 1 − z + 2 k π i ) , k ∈ ℤ |
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coth − 1 z = 1 2 ln ( z + 1 z − 1 + 2 k π i ) , k ∈ ℤ |
