Die normierte Planckfunktion
                         ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀

      Die spektrale Energieverteilung innerhalb eines 'schwarzen'
      Hohlraumes, dessen Wände die Temperatur T haben, wird durch das
      Plancksche Strahlungsgesetz beschrieben.  Die Strahlungsleistung
      dP, die pro Flächenelement dA in das Raumwinkelelement dΩ
      (orthogonal zu dA) im Wellenlängenintervall dl als Funktion der
      Vakuumwellenlänge l abgestrahlt wird, ist nach Planck:

                                                  ┌        ┐
               2∙h∙c²      1                      │   W    │
          dP = ──────∙────────────  , Dimension:  │────────│ ,       (1)
                l^5      ┌ c2┐                    │m^3∙rad²│
                      exp│───│ - 1                └        ┘
                         └T∙l┘

      mit der zur Abkürzung eingeführten Größe:

                h∙c
          c2 := ─── = 1.43877485e-2 [m∙K]   .                        (2)
                 k

      Die in (1) bzw. (2) vorkommenden Größen sind im Einzelnen:

          h := 6.6260693e-34  [J∙s]    Plancksches Wirkungsquantum,
          c := 2.99792458e+8  [m/s]    Vakuum-Lichtgeschwindigkeit,
          k := 1.3806505e-23  [J/K]    Boltzmann-Konstante,
          T :=                [K]      absolute Temperatur,
          l :=                [m]      Wellenlänge .

      Gefragt ist nun die über alle Wellenlängen integrierte Leistung.
      Wie sich zeigen wird, ist (1) geschlossen über l integrierbar.
      Dazu benutzen wir die Integraldarstellung der Riemannschen
      Zeta-Funktion Z(s):

                        ∞
                    1   ⌠    u^(s-1)
          Z(s) := ────∙ │ ────────────∙du   für Re(s) > 1.           (3)
                  Γ(s)  ⌡  exp(u) - 1
                       u=0

      Insbesondere ist:

                 π^4
          Z(4) = ──── , so daß
                  90

           ∞
           ⌠      u^3                      π^4       π^4
           │ ────────────∙du = Z(4)∙Γ(4) = ────∙3! = ──── .          (4)
           ⌡  exp(u) - 1                    90        15
          u=0

      Nun soll die Funktion (1) so normiert werden, daß das Integral
      über alle Wellenlängen [0,∞] den Wert 1 ergibt. Das Integral von
      (1) über alle Wellenlängen l ist:

                      ∞
                      ⌠
                      │                        ┌       ┐
                      │         dl             │   W   │
          dP = 2∙h∙c²∙│ ──────────────────  ,  │───────│ ,           (5)
                      │     ┌   ┌ c2┐    ┐     │m²∙rad²│
                      │ l^5∙│exp│───│ - 1│     └       ┘
                      ⌡     └   └T∙l┘    ┘
                     l=0

      und mittels der Substitution:

           c2                c2            -c2
          ─── = u  bzw. l = ───  und  dl = ────∙du                   (6)
          T∙l               T∙u            T∙u²

      geht es über in:

                      ┌  ┐5  0
                      │ T│   ⌠    -u^5      c2
          dP = 2∙h∙c²∙│──│ ∙ │ ───────────∙────∙du ,
                      │c2│   ⌡ exp(u) - 1  T∙u²
                      └  ┘  u=∞

                      ┌  ┐4  ∞
                      │ T│   ⌠     u^3
          dP = 2∙h∙c²∙│──│ ∙ │ ───────────∙du  ,
                      │c2│   ⌡ exp(u) - 1
                      └  ┘  u=0

      und wenn man (4) einsetzt:

                          ┌  ┐4      ┌       ┐
               2∙h∙c²∙π^4 │ T│       │   W   │
          dP = ──────────∙│──│     , │───────│  .                    (7)
                   15     │c2│       │m²∙rad²│
                          └  ┘       └       ┘

      Um die Leistung dP zu erhalten, die vom Flächenelement dA
      einseitig in den Halbraum (Raumwinkel 2∙π) abgestrahlt wird, muß
      man noch über dΩ integrieren, das ergibt den Faktor π, so daß man
      erhält:

                          ┌  ┐4    ┌    ┐
               2∙h∙c²∙π^5 │ T│     │ W  │
          dP = ──────────∙│──│   , │────│    ,                       (8)
                   15     │c2│     │ m² │
                          └  ┘     └    ┘

      und ersetzt man in (8) c2 nach (2), so folgt:

                                                            ┌    ┐
               2∙h∙c²∙π^5∙k^4       2∙π^5∙k^4               │ W  │
          dP = ──────────────∙T^4 = ─────────∙T^4 = σ∙T^4 , │────│.  (9)
               15∙h^4∙c^4           15∙h^3∙c²               │ m² │
                                                            └    ┘

      Das ist das Stefan-Boltzmann-Gesetz mit der sogenannten
      Stefan-Boltzmann-Kontanten σ:

              2∙π^5∙k^4
          σ = ───────── = 5.67040124e-8 [W/(m²∙K^4)]  .             (10)
              15∙h^3∙c²

      Kommen wir nach diesen Vorbemerkungen nun zur Bestimmung der
      'normierten' Plankfunktion.

      Dividiert man die Funktion (1) durch die Größe (7), so erhalten
      wir eine 'normierte' Funktion dPn, deren Integral über l von 0 bis
      ∞ den Wert 1 ergibt:

                   ┌   ┐4                        ┌   ┐
                   │ c2│          1              │ 1 │
          dPn = 15∙│───│ ∙──────────────────   , │───│   .          (11)
                   │π∙T│      ┌   ┌ c2┐    ┐     │ m │
                   └   ┘  l^5∙│exp│───│ - 1│     └   ┘
                              └   └T∙l┘    ┘

      Hat man z. B.  Meßwerte yi der Dimension [W/(m²∙m)] für
      verschiedene Wellenlängen li vorliegen, so würde man zum Bestimmen
      der Funktion durch Ausgleichen ansetzen:

                      ┌    ┐4
                      │ c2 │            1
          yi  = A1∙15∙│────│ ∙─────────────────────  ,              (12)
                      │π∙A2│       ┌   ┌ c2  ┐    ┐
                      └    ┘  li^5∙│exp│─────│ - 1│
                                   └   └A2∙li┘    ┘

      mit den zu bestimmenden Parametern A1 und A2.  A1 stellt dann die
      (integrierte) Gesamtleistung pro Fläche über alle Wellenlängen
      dar, A2 ist die Temperatur.

      Ein andere Ansatz berücksichtigt zusätzlich einen (durch
      fehlerhafte Meßtechnik begründeten) Untergrund:

                           ┌    ┐4
                           │ c2 │            1
          yi  = A1 + A2∙15∙│────│ ∙─────────────────────  ,         (13)
                           │π∙A3│       ┌   ┌ c2  ┐    ┐
                           └    ┘  li^5∙│exp│─────│ - 1│
                                        └   └A3∙li┘    ┘

      mit A1: Untergrund,
          A2: Gesamtleistung pro Fläche
          A3: Temperatur.


                                    Anhang
                                    ▀▀▀▀▀▀

      Die partiellen Ableitungen nach den zu bestimmenden Parametern A1
      und A2, Fall 1, bzw. A1,A2,A3, Fall 2, sollen hier notiert werden.

      Fall 1, (12):

                   ┌    ┐4
          dyi      │ c2 │           1
          ─── = 15∙│────│ ∙─────────────────────   ,
          dA1      │π∙A2│       ┌   ┌ c2  ┐    ┐
                   └    ┘  li^5∙│exp│─────│ - 1│
                                └   └A2∙li┘    ┘

                       ┌                                     ┐
                       │                                     │
          dyi   A1 dyi │       c2      ┌ c2  ┐    1          │
          ─── = ──∙───∙│ -4 + ─────∙exp│─────│∙──────────────│ .
          dA2   A2 dA1 │      A2∙li    └A2∙li┘    ┌ c2  ┐    │
                       │                       exp│─────│ - 1│
                       └                          └A2∙li┘    ┘

      Fall 2, (13):

          dyi
          ─── = 1  ,
          dA1

                   ┌    ┐4
          dyi      │ c2 │            1
          ─── = 15∙│────│ ∙─────────────────────   ,
          dA2      │π∙A3│       ┌   ┌ c2  ┐    ┐
                   └    ┘  li^5∙│exp│─────│ - 1│
                                └   └A3∙li┘    ┘

                       ┌                                     ┐
                       │                                     │
          dyi   A2 dyi │       c2      ┌ c2  ┐       1       │
          ─── = ──∙───∙│ -4 + ─────∙exp│─────│∙──────────────│ .
          dA3   A3 dA1 │      A3∙li    └A3∙li┘    ┌ c2  ┐    │
                       │                       exp│─────│ - 1│
                       └                          └A3∙li┘    ┘

                                    * * *


      Numerisches Beispiel mit zwei Parametern für die Sonnenstrahlung
      ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀


      Daten von RReDC:
      http://rredc.nrel.gov/solar/spectra/am0/ASTM2000.html.

      ------------------------------------------------------------------
      Equations N = 880, Unknowns  M = 2

      Data reading:   0.00sec

          Iteration       AM0(1)       AM0(2)        delta

                  0 +1.4000e+003 +5.7000e+003 +0.0000e+000
                  1 +1.4470e+003 +5.4124e+003 +8.5644e-002
                  2 +1.4433e+003 +5.4517e+003 +9.7929e-003
                  3 +1.4448e+003 +5.4465e+003 +2.0128e-003
                  4 +1.4447e+003 +5.4472e+003 +2.4896e-004
                  5 +1.4447e+003 +5.4471e+003 +3.2517e-005
                  6 +1.4447e+003 +5.4471e+003 +4.2039e-006

      Observations  Positions  Weights  Mean obs. Mean errors  Residuals

         Y [W/m^3]      X [m]        W YM [W/m^3] FYM [W/m^3]  V [W/m^3]

        +5.0000e+6 +1.9950e-7     +1.0 +6.0934e+7  +1.5910e+6 -5.5934e+7
        +7.0000e+6 +2.0050e-7     +1.0 +6.3486e+7  +1.6446e+6 -5.6486e+7
        +7.0000e+6 +2.0150e-7     +1.0 +6.6111e+7  +1.6993e+6 -5.9111e+7
        +8.0000e+6 +2.0250e-7     +1.0 +6.8807e+7  +1.7548e+6 -6.0807e+7
        +9.0000e+6 +2.0350e-7     +1.0 +7.1577e+7  +1.8113e+6 -6.2577e+7
        +9.0000e+6 +2.0450e-7     +1.0 +7.4421e+7  +1.8687e+6 -6.5421e+7
        +1.0000e+7 +2.0550e-7     +1.0 +7.7339e+7  +1.9269e+6 -6.7339e+7
        +1.0000e+7 +2.0650e-7     +1.0 +8.0332e+7  +1.9861e+6 -7.0332e+7
        +1.1000e+7 +2.0750e-7     +1.0 +8.3401e+7  +2.0461e+6 -7.2401e+7
        +1.5000e+7 +2.0850e-7     +1.0 +8.6546e+7  +2.1069e+6 -7.1546e+7
        +2.4000e+7 +2.0950e-7     +1.0 +8.9767e+7  +2.1686e+6 -6.5767e+7
        +2.8000e+7 +2.1050e-7     +1.0 +9.3066e+7  +2.2311e+6 -6.5066e+7
        +3.4000e+7 +2.1150e-7     +1.0 +9.6441e+7  +2.2944e+6 -6.2441e+7
        +3.0000e+7 +2.1250e-7     +1.0 +9.9895e+7  +2.3585e+6 -6.9895e+7
        +3.2000e+7 +2.1350e-7     +1.0 +1.0343e+8  +2.4233e+6 -7.1427e+7
        +4.1000e+7 +2.1450e-7     +1.0 +1.0704e+8  +2.4889e+6 -6.6038e+7
        +3.7000e+7 +2.1550e-7     +1.0 +1.1073e+8  +2.5552e+6 -7.3727e+7
        +3.4000e+7 +2.1650e-7     +1.0 +1.1450e+8  +2.6222e+6 -8.0495e+7
        +3.6000e+7 +2.1750e-7     +1.0 +1.1834e+8  +2.6899e+6 -8.2343e+7
        +4.5000e+7 +2.1850e-7     +1.0 +1.2227e+8  +2.7582e+6 -7.7269e+7
        +4.8000e+7 +2.1950e-7     +1.0 +1.2628e+8  +2.8272e+6 -7.8276e+7
        +4.8000e+7 +2.2050e-7     +1.0 +1.3036e+8  +2.8968e+6 -8.2362e+7
        +3.9000e+7 +2.2150e-7     +1.0 +1.3453e+8  +2.9671e+6 -9.5528e+7
        +5.1000e+7 +2.2250e-7     +1.0 +1.3877e+8  +3.0378e+6 -8.7774e+7
        +6.6000e+7 +2.2350e-7     +1.0 +1.4310e+8  +3.1092e+6 -7.7099e+7
        +5.8000e+7 +2.2450e-7     +1.0 +1.4750e+8  +3.1811e+6 -8.9505e+7
        +5.4000e+7 +2.2550e-7     +1.0 +1.5199e+8  +3.2535e+6 -9.7990e+7
        +4.1000e+7 +2.2650e-7     +1.0 +1.5655e+8  +3.3263e+6 -1.1555e+8
        +4.1000e+7 +2.2750e-7     +1.0 +1.6120e+8  +3.3997e+6 -1.2020e+8
        +5.4000e+7 +2.2850e-7     +1.0 +1.6592e+8  +3.4734e+6 -1.1192e+8
        +4.8000e+7 +2.2950e-7     +1.0 +1.7073e+8  +3.5476e+6 -1.2273e+8
        +5.6000e+7 +2.3050e-7     +1.0 +1.7561e+8  +3.6222e+6 -1.1961e+8
        +5.0000e+7 +2.3150e-7     +1.0 +1.8057e+8  +3.6971e+6 -1.3057e+8
        +5.5000e+7 +2.3250e-7     +1.0 +1.8561e+8  +3.7724e+6 -1.3061e+8
        +4.6000e+7 +2.3350e-7     +1.0 +1.9072e+8  +3.8480e+6 -1.4472e+8
        +3.9000e+7 +2.3450e-7     +1.0 +1.9592e+8  +3.9239e+6 -1.5692e+8
        +5.7000e+7 +2.3550e-7     +1.0 +2.0119e+8  +4.0001e+6 -1.4419e+8
        +4.9000e+7 +2.3650e-7     +1.0 +2.0654e+8  +4.0765e+6 -1.5754e+8
        +5.3000e+7 +2.3750e-7     +1.0 +2.1196e+8  +4.1531e+6 -1.5896e+8
        +4.2000e+7 +2.3850e-7     +1.0 +2.1746e+8  +4.2300e+6 -1.7546e+8
        +4.6000e+7 +2.3950e-7     +1.0 +2.2304e+8  +4.3069e+6 -1.7704e+8
        +4.3000e+7 +2.4050e-7     +1.0 +2.2869e+8  +4.3841e+6 -1.8569e+8
        +5.2000e+7 +2.4150e-7     +1.0 +2.3442e+8  +4.4613e+6 -1.8242e+8
        +7.2000e+7 +2.4250e-7     +1.0 +2.4022e+8  +4.5387e+6 -1.6822e+8
        +6.5000e+7 +2.4350e-7     +1.0 +2.4609e+8  +4.6161e+6 -1.8109e+8
        +6.2000e+7 +2.4450e-7     +1.0 +2.5204e+8  +4.6936e+6 -1.9004e+8
        +5.1000e+7 +2.4550e-7     +1.0 +2.5806e+8  +4.7711e+6 -2.0706e+8
        +5.1000e+7 +2.4650e-7     +1.0 +2.6415e+8  +4.8486e+6 -2.1315e+8
        +5.7000e+7 +2.4750e-7     +1.0 +2.7031e+8  +4.9261e+6 -2.1331e+8
        +4.5000e+7 +2.4850e-7     +1.0 +2.7654e+8  +5.0035e+6 -2.3154e+8
        +5.8000e+7 +2.4950e-7     +1.0 +2.8284e+8  +5.0809e+6 -2.2484e+8
        +5.9000e+7 +2.5050e-7     +1.0 +2.8920e+8  +5.1581e+6 -2.3020e+8
        +4.7000e+7 +2.5150e-7     +1.0 +2.9564e+8  +5.2353e+6 -2.4864e+8
        +4.4000e+7 +2.5250e-7     +1.0 +3.0214e+8  +5.3123e+6 -2.5814e+8
        +5.5000e+7 +2.5350e-7     +1.0 +3.0871e+8  +5.3891e+6 -2.5371e+8
        +6.1000e+7 +2.5450e-7     +1.0 +3.1534e+8  +5.4657e+6 -2.5434e+8
        +8.9000e+7 +2.5550e-7     +1.0 +3.2204e+8  +5.5422e+6 -2.3304e+8
        +1.0700e+8 +2.5650e-7     +1.0 +3.2880e+8  +5.6184e+6 -2.2180e+8
        +1.2900e+8 +2.5750e-7     +1.0 +3.3562e+8  +5.6944e+6 -2.0662e+8
        +1.3400e+8 +2.5850e-7     +1.0 +3.4250e+8  +5.7700e+6 -2.0850e+8
        +1.0800e+8 +2.5950e-7     +1.0 +3.4945e+8  +5.8454e+6 -2.4145e+8
        +1.0200e+8 +2.6050e-7     +1.0 +3.5645e+8  +5.9205e+6 -2.5445e+8
        +1.0300e+8 +2.6150e-7     +1.0 +3.6351e+8  +5.9952e+6 -2.6051e+8
        +1.2100e+8 +2.6250e-7     +1.0 +3.7063e+8  +6.0696e+6 -2.4963e+8
        +1.7500e+8 +2.6350e-7     +1.0 +3.7781e+8  +6.1436e+6 -2.0281e+8
        +2.7400e+8 +2.6450e-7     +1.0 +3.8504e+8  +6.2173e+6 -1.1104e+8
        +2.8000e+8 +2.6550e-7     +1.0 +3.9233e+8  +6.2905e+6 -1.1233e+8
        +2.6000e+8 +2.6650e-7     +1.0 +3.9967e+8  +6.3632e+6 -1.3967e+8
        +2.7000e+8 +2.6750e-7     +1.0 +4.0706e+8  +6.4355e+6 -1.3706e+8
        +2.6000e+8 +2.6850e-7     +1.0 +4.1450e+8  +6.5074e+6 -1.5450e+8
        +2.5200e+8 +2.6950e-7     +1.0 +4.2200e+8  +6.5787e+6 -1.7000e+8
        +2.9300e+8 +2.7050e-7     +1.0 +4.2954e+8  +6.6496e+6 -1.3654e+8
        +2.3200e+8 +2.7150e-7     +1.0 +4.3713e+8  +6.7199e+6 -2.0513e+8
        +2.1500e+8 +2.7250e-7     +1.0 +4.4477e+8  +6.7897e+6 -2.2977e+8
        +2.0400e+8 +2.7350e-7     +1.0 +4.5245e+8  +6.8590e+6 -2.4845e+8
        +1.3700e+8 +2.7450e-7     +1.0 +4.6018e+8  +6.9276e+6 -3.2318e+8
        +2.0000e+8 +2.7550e-7     +1.0 +4.6795e+8  +6.9957e+6 -2.6795e+8
        +2.5800e+8 +2.7650e-7     +1.0 +4.7577e+8  +7.0632e+6 -2.1777e+8
        +2.4000e+8 +2.7750e-7     +1.0 +4.8362e+8  +7.1301e+6 -2.4362e+8
        +1.6600e+8 +2.7850e-7     +1.0 +4.9152e+8  +7.1963e+6 -3.2552e+8
        +8.9000e+7 +2.7950e-7     +1.0 +4.9945e+8  +7.2619e+6 -4.1045e+8
        +1.1200e+8 +2.8050e-7     +1.0 +5.0742e+8  +7.3269e+6 -3.9542e+8
        +2.3100e+8 +2.8150e-7     +1.0 +5.1543e+8  +7.3912e+6 -2.8443e+8
        +3.0700e+8 +2.8250e-7     +1.0 +5.2348e+8  +7.4548e+6 -2.1648e+8
        +3.3000e+8 +2.8350e-7     +1.0 +5.3156e+8  +7.5177e+6 -2.0156e+8
        +2.4400e+8 +2.8450e-7     +1.0 +5.3967e+8  +7.5799e+6 -2.9567e+8
        +1.4100e+8 +2.8550e-7     +1.0 +5.4781e+8  +7.6413e+6 -4.0681e+8
        +3.2000e+8 +2.8650e-7     +1.0 +5.5599e+8  +7.7021e+6 -2.3599e+8
        +3.7100e+8 +2.8750e-7     +1.0 +5.6419e+8  +7.7621e+6 -1.9319e+8
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        +5.7000e+7 +2.4230e-6     +1.0 +6.5663e+7  +7.6157e+5 -8.6627e+6
        +5.1000e+7 +2.4430e-6     +1.0 +6.3873e+7  +7.4193e+5 -1.2873e+7
        +5.3000e+7 +2.4680e-6     +1.0 +6.1721e+7  +7.1824e+5 -8.7206e+6
        +5.4000e+7 +2.4930e-6     +1.0 +5.9657e+7  +6.9546e+5 -5.6565e+6
        +4.7000e+7 +2.5180e-6     +1.0 +5.7677e+7  +6.7356e+5 -1.0677e+7


      Unknowns AM,
      Std. deviation FAM of AM,
      Std. deviation FY of single measurement Y[i]:

       +1.4447e+003 +5.4471e+003

       ±8.0543e+000 ±1.7099e+001

       ±1.5305e+008

      Correlations of AM (diagonal = sample variances):

       +6.4871e+001

       -5.8951e-001 +2.9238e+002

      Calculation:   0.11sec
      -------------------------------------------------------------------
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Daten von RReDC.