Étiquette : engineering-function

Its computes the principal value of the base-10 logarithm of a complex number, returning the result in rectangular form (x + yi). This is the complex analog of the standard LOG10() function.

Syntax
IMLOG10(complex_number)

Argument

• complex_number (required)
  A complex number in either:
  • « x+yi » format (mathematical convention)
  • « x+yj » format (engineering convention)

Technical Background
For a complex number z = x + yi:

IMLOG10(z) = (log₁₀e)·IMLN(z) ≈ 0.434294482·IMLN(z)

Where:

• IMLN(z) is the complex natural logarithm
• log₁₀e ≈ 0.434294481903252

Example

=IMLOG10(« 3-4i »)  // Returns « 0.698970004-0.402719196i »

Additional Examples

=IMLOG10(« 1+i »)      // Returns « 0.150514998+0.341094088i »

=IMLOG10(« -100 »)     // Returns « 2+1.364376354i »

=IMLOG10(« 0+10i »)    // Returns « 1+0.682188177i »

=IMLOG10(« 1000 »)     // Returns « 3 » (standard real logarithm)

Key Features

• Returns the principal value (-π < Im ≤ π)
• Maintains 9 decimal place precision
• Handles all valid complex number formats
• Consistent with real LOG10() for positive reals

Its computes the principal value of the base-2 logarithm of a complex number, returning the result in rectangular form (x + yi). This is the complex analog of the standard LOG() function with base 2.

Syntax
IMLOG2(complex_number)

Argument

• complex_number (required)
  A complex number in either:
  • « x+yi » format (mathematical convention)
  • « x+yj » format (engineering convention)

Technical Background
For a complex number z = x + yi:

IMLOG2(z) = (log₂e)·IMLN(z) ≈ 1.442695041·IMLN(z)

Where:

• IMLN(z) is the complex natural logarithm
• log₂e ≈ 1.4426950408889634

Example

=IMLOG2(« 3-4i »)  // Returns « 2.321928095-1.337804361i »

Additional Examples

=IMLOG2(« 1+i »)      // Returns « 0.5+1.133090035i »

=IMLOG2(« -8 »)       // Returns « 3+4.532360141i »

=IMLOG2(« 0+16i »)    // Returns « 4+2.266180071i »

=IMLOG2(« 1024 »)     // Returns « 10 » (standard real base-2 log)

Key Features

• Returns the principal value (-π < Im ≤ π)
• Maintains 9 decimal place precision
• Handles all valid complex number formats
• Consistent with real base-2 logarithm for positive reals

Error Conditions

• Returns #NUM! for:
  • Invalid complex number format
  • Non-numeric components
  • Zero magnitude (0+0i)

Its computes the principal value of the natural logarithm of a complex number, returning the result in rectangular form (x + yi). This is the complex analog of the standard LN() function.

Syntax
IMLN(complex_number)

Argument

• complex_number (required)
  A complex number in either:
  • « x+yi » format (mathematical convention)
  • « x+yj » format (engineering convention)

Technical Background
For a complex number z = x + yi:

IMLN(z) = ln|z| + i·arg(z)

Where:

• |z| = √(x² + y²) (magnitude)
• arg(z) = atan2(y,x) (phase angle in [-π, π])

Example

=IMLN(« 3-4i »)  // Returns « 1.60943791-0.927295218i »

Additional Examples

=IMLN(« 1+i »)      // Returns « 0.34657359+0.785398163i »

=IMLN(« -1 »)       // Returns « 0+3.141592654i » (ln(-1) = iπ)

=IMLN(« 0+1i »)     // Returns « 0+1.570796327i » (ln(i) = iπ/2)

=IMLN(« 2.718281828459 ») // Returns « 1 » (ln(e) = 1)

Key Features

• Returns the principal value (-π < Im ≤ π)
• Maintains 9 decimal place precision
• Handles all valid complex number formats
• Preserves input’s imaginary unit convention

Error Conditions

• Returns #NUM! for:
  • Invalid complex number format
  • Non-numeric components
  • Zero magnitude (0+0i)

Its computes the exponential of a complex number, returning the result in rectangular form (x + yi). This is the complex analog of the standard EXP() function.

Syntax
IMEXP(complex_number)

Argument

• complex_number (required)
  A complex exponent in either:
  • « x+yi » format (mathematical convention)
  • « x+yj » format (engineering convention)

Technical Background
For a complex number z = x + yi:

IMEXP(z) = e^z = e^x · (cos y + i·sin y)

This implementation of Euler’s formula:

1. Separates into magnitude (eˣ) and phase (eⁱʸ) components
2. Converts the imaginary exponent using Euler’s identity

Example

=IMEXP(« 3-4i »)  // Returns « -13.1287831+15.2007845i »

Additional Examples

=IMEXP(« 1+i »)      // Returns « 1.46869394+2.28735529i »

=IMEXP(« 0+πi »)     // Returns « -1 » (Euler’s identity)

=IMEXP(« 2+0i »)     // Returns « 7.3890561 » (matches real EXP(2))

Key Features

• Maintains 8 decimal place precision
• Preserves input’s imaginary unit convention
• Handles all valid complex number formats
• Periodic in imaginary dimension (period 2π)

Error Conditions

• Returns #NUM! for:
  • Invalid complex number format
  • Non-numeric components
  • Missing imaginary unit when required

Usage Notes

1. For pure real numbers, equivalent to EXP()
2. For pure imaginary numbers, reduces to Euler’s formula
3. Essential for:
   • Complex differential equations
   • Fourier transforms
   • Quantum mechanical wavefunctions

Its calculates the quotient of two complex numbers in rectangular form (x + yi or x + yj).

Syntax
IMDIV(complex_number1; complex_number2)

Arguments

• complex_number1 (required)
  The complex numerator (dividend) in « x+yi » or « x+yj » format
• complex_number2 (required)
  The complex denominator (divisor) in matching format

Technical Background
For complex numbers z₁ = a + bi and z₂ = c + di:

• Division is performed by rationalizing the denominator:

z₁/z₂ = [(ac + bd) + i(bc – ad)] / (c² + d²)

• Equivalent to multiplying numerator by complex conjugate of denominator

Example

=IMDIV(« 3-4i », »-7-24i »)  // Returns « 0.12+0.16i »

Additional Examples

=IMDIV(« 1+i », »1-i »)       // Returns « i »

=IMDIV(« 4″, »2i »)          // Returns « -2i »

=IMDIV(« 3+4i », »3+4i »)     // Returns « 1 »

=IMDIV(« 10″, »2+3i »)       // Returns « 1.538461538-2.307692308i »

Key Features

• Returns result in standard complex form
• Maintains input’s imaginary unit convention
• Handles all valid complex number formats
• Properly handles division by pure real/imaginary numbers

Its calculates the cosine of a complex number in rectangular form (x + yi or x + yj).

Syntax
IMCOS(complex_number)

Argument

• complex_number (required)
  A text string representing a complex number in either:
  • « x+yi » format (mathematical convention)
  • « x+yj » format (engineering convention)

Technical Background
For a complex number z = x + yi:

• Complex cosine is calculated as:
  cos(z) = cos(x)cosh(y) – i·sin(x)sinh(y)
• Uses Euler’s formula extension to complex plane
• Combines trigonometric and hyperbolic functions

Example

=IMCOS(« 3-4i »)  // Returns « -27.0349456+3.85115334i »

Additional Examples

=IMCOS(« 1+i »)     // Returns « 0.833730025-0.988897706i »

=IMCOS(« 0+πi »)    // Returns « 11.59195328 » (real number)

=IMCOS(« π/2-i »)   // Returns « 1.543080635i » (pure imaginary)

Key Features

• Returns result in standard complex form
• Preserves input’s imaginary unit convention
• Handles all complex number formats
• Maintains periodicity properties in complex plane

Error Conditions

• Returns #NUM! for:
  • Invalid complex number format
  • Non-numeric components
  • Missing imaginary unit when needed

Usage Notes

1. For real numbers (y=0), reduces to standard cosine
2. For pure imaginary numbers (x=0), becomes hyperbolic cosine:
   cos(iy) = cosh(y)
3. Essential for:
   • Complex analysis
   • Wave propagation modeling
   • Quantum mechanical calculations

Its returns the complex conjugate of a complex number in rectangular form (x + yi or x + yj).

Syntax
IMCONJUGATE(complex_number)

Argument

• complex_number (required)
  A text string representing a complex number in either:
  • « x+yi » format (mathematical convention)
  • « x+yj » format (engineering convention)

Technical Background
For a complex number z = x + yi:

• The complex conjugate is z̅ = x – yi
• Geometric interpretation: Reflection across the real axis
• Key property: z·z̅ = x² + y² = |z|²

Example

=IMCONJUGATE(« 3-4i »)  // Returns « 3+4i »

Additional Examples

=IMCONJUGATE(« 5+12j »)   // Returns « 5-12j »

=IMCONJUGATE(« -1+i »)    // Returns « -1-i »

=IMCONJUGATE(« 7 »)       // Returns « 7 » (real numbers are unchanged)

=IMCONJUGATE(« 0-3i »)    // Returns « 0+3i »

Key Features

• Preserves the original imaginary unit (i/j)
• Returns result as text string
• Handles all complex number formats
• Identity operation for real numbers

Error Conditions

• Returns #NUM! for:
  • Invalid complex number format
  • Non-numeric components
  • Missing imaginary unit when needed

Usage Notes

1. Essential for division of complex numbers
2. Used in quantum mechanics (bra-ket notation)
3. Important for calculating magnitudes:

=IMPRODUCT(« 3-4i », IMCONJUGATE(« 3-4i »))  // Returns « 25 » (3² + 4²)

This function calculates the argument (phase angle Φ) in radians of a complex number given in rectangular form (x + yi or x + yj).

Syntax
IMARGUMENT(complex_number)

Argument

• complex_number (required)
  A text string representing a complex number in either:
  • « x+yi » format (mathematical convention)
  • « x+yj » format (engineering convention)

Technical Background
For a complex number z = x + yi:

• The argument Φ is the angle in the complex plane
• Calculated as: Φ = atan2(y, x)
• Range: -π < Φ ≤ π radians
• Relates to trigonometric form: |z|·(cosΦ + i·sinΦ)

Example

=IMARGUMENT(« 3-4i »)  // Returns -0.927295218 radians

Additional Examples

=IMARGUMENT(« 1+i »)     // Returns 0.785398163 (π/4 radians)

=IMARGUMENT(« -1-i »)    // Returns -2.35619449 (-3π/4 radians)

=IMARGUMENT(« 0+1j »)    // Returns 1.570796327 (π/2 radians)

=IMARGUMENT(« 5 »)       // Returns 0 (pure real number)

Key Features

• Returns angle in radians (-π to π)
• Supports both « i » and « j » notation
• Handles all four quadrants of complex plane
• Returns 0 for positive real numbers
• Returns π for negative real numbers

Error Conditions

• Returns #NUM! for:
  • Invalid complex number format
  • Non-numeric components
  • Empty string

Usage Notes

1. To convert result to degrees, use DEGREES() function
2. For zero (0+0i), returns 0 by convention
3. Angle sign follows standard mathematical convention:
   • Positive for upper half-plane
   • Negative for lower half-plane

Its extracts the imaginary coefficient (y) from a complex number in the form x + yi or x + yj.

Syntax
IMAGINARY(complex_number)

Argument

• complex_number (required)
  A text string representing a complex number in either:
  • « x+yi » format (mathematical convention)
  • « x+yj » format (engineering convention)
  • Examples: « 3-4i », « 5+2j », « -1+7i »

Technical Background
For a complex number z = a + bi:

• a is the real part (extracted with IMREAL())
• b is the imaginary part (extracted with IMAGINARY())
• i (or j) is the imaginary unit (√-1)

Example

=IMAGINARY(« 3-4i »)  // Returns -4

Key Features

• Supports both « i » and « j » notation
• Returns a real number value
• Handles positive/negative coefficients
• Returns 0 for pure real numbers

Error Conditions

• Returns #NUM! for:
  • Invalid complex number format
  • Missing imaginary unit when imaginary part exists
  • Non-numeric components

Usage Notes

1. Input must be text string in proper complex format
2. The sign of the imaginary part is preserved
3. For engineering applications, use « j » notation
4. Combine with IMREAL() to separate complex components

Its calculates the absolute value (modulus) of a complex number. The complex number can be provided in either « x+yi » or « x+yj » format.

Syntax
IMABS(complex_number)

Argument

• complex_number (required)
  A complex number in text format with:
  • Real and imaginary components
  • Either « i » or « j » as the imaginary unit
  • Examples: « 3+4i », « 5-2j », « -1+7i »

Technical Background
The absolute value of a complex number a + bi is calculated as:
√(a² + b²)

Example

=IMABS(« 3-4i »)  // Returns 5

Additional Examples

=IMABS(« 5+12i »)    // Returns 13

=IMABS(« 1+i »)      // Returns 1.414213562 (√2)

=IMABS(« 0+1j »)     // Returns 1

=IMABS(« -3-4j »)    // Returns 5

Key Features

• Accepts both « i » and « j » notation
• Handles positive and negative components
• Returns a real number value
• Follows standard complex number mathematics

Error Conditions

• Returns #NUM! for:
  • Invalid complex number format
  • Missing imaginary unit
  • Non-numeric components

Usage Notes

1. Input must be text string in proper complex format
2. Components can be integers or decimals
3. Result is always a non-negative real number